How To Understand The Theory Of Inequlities

Transcription

1 Ostrowski Type Inequlities nd Applictions in Numericl Integrtion Edited By: Sever S Drgomir nd Themistocles M Rssis SS Drgomir) School nd Communictions nd Informtics, Victori University of Technology, PO Box 448, Melbourne City, MC 800, Victori, Austrli E-mil ddress, SS Drgomir: sever@mtildvueduu URL: TM Rssis) Deprtment of Mthemtics, Ntionl Technicl University of Athens, Zogrfou Cmpus, 5780 Athens, Greece

2 99 Mthemtics Subject Clssifiction Primry 6D5, 6D0; Secondry 4A55, 4A99 Abstrct Ostrowski type inequlities for univrite nd multivrite rel functions nd their nturl pplictions for numericl qudrtures re presented

3 Contents Prefce v Chpter Generlistions of Ostrowski Inequlity nd Applictions Introduction Generlistions for Functions of Bounded Vrition 3 3 Generlistions for Functions whose Derivtives re in L 5 4 Generlistion for Functions whose Derivtives re in L p 8 5 Generlistions in Terms of L norm 4 Bibliogrphy 53 Chpter Integrl Inequlities for n Times Differentible Mppings 55 Introduction 55 Integrl Identities 56 3 Integrl Inequlities 64 4 The Convergence of Generl Qudrture Formul 7 5 Grüss Type Inequlities 75 6 Some Prticulr Integrl Inequlities 80 7 Applictions for Numericl Integrtion 04 8 Concluding Remrks 8 Bibliogrphy 9 Chpter 3 Three Point Qudrture Rules 3 Introduction 3 Bounds Involving t most First Derivtive 3 33 Bounds for n Time Differentible Functions 86 Bibliogrphy Chpter 4 Product Brnches of Peno Kernels nd Numericl Integrtion 5 4 Introduction 5 4 Fundmentl Results 7 43 Simpson Type Formule 4 44 Perturbed Results 6 45 More Perturbed Results Using Seminorms Concluding Remrks 4 Bibliogrphy 43 Chpter 5 Ostrowski Type Inequlities for Multiple Integrls 45 5 Introduction 45 iii

4 CONTENTS iv 5 An Ostrowski Type Inequlity for Double Integrls Other Ostrowski Type Inequlities Ostrowski s Inequlity for Hölder Type Functions 73 Bibliogrphy 8 Chpter 6 Some Results for Double Integrls Bsed on n Ostrowski Type Inequlity 83 6 Introduction 83 6 The One Dimensionl Ostrowski Inequlity Mpping Whose First Derivtives Belong to L, b) Numericl Results Appliction For Cubture Formule Mpping Whose First Derivtives Belong to L p, b) Appliction For Cubture Formule Mppings Whose First Derivtives Belong to L, b) Integrl Identities Some Integrl Inequlities Applictions to Numericl Integrtion 3 Bibliogrphy 35 Chpter 7 Product Inequlities nd Weighted Qudrture 37 7 Introduction 37 7 Weight Functions Weighted Interior Point Integrl Inequlities Weighted Boundry Point Lobtto) Integrl Inequlities Weighted Three Point Integrl Inequlities 339 Bibliogrphy 35 Chpter 8 Some Inequlities for Riemnn-Stieltjes Integrl Introduction Some Trpezoid Like Inequlities for Riemnn-Stieltjes Integrl Inequlities of Ostrowski Type for the Riemnn-Stieltjes Integrl Some Inequlities of Grüss Type for Riemnn-Stieltjes Integrl 39 Bibliogrphy 40

5 v Prefce It ws noted in the prefce of the book Inequlities Involving Functions nd Their Integrls nd Derivtives, Kluwer Acdemic Publishers, 99, by DS Mitrinović, JE Pečrić nd AM Fink; since the writing of the clssicl book by Hrdy, Littlewood nd Poly 934), the subject of differentil nd integrl inequlities hs grown by bout 800% Ten yers on, we cn confidently ssert tht this growth will increse even more significntly Twenty pges of Chpter XV in the bove mentioned book re devoted to integrl inequlities involving functions with bounded derivtives, or, Ostrowski type inequlities This is now itself specil domin of the Theory of Inequlities with mny powerful results nd lrge number of pplictions in Numericl Integrtion, Probbility Theory nd Sttistics, Informtion Theory nd Integrl Opertor Theory The min im of this present book, jointly written by the members of the Victori University node of RGMIA Reserch Group in Mthemticl Inequlities nd Applictions, is to present selected number of results on Ostrowski type inequlities Results for univrite nd multivrite rel functions nd their nturl pplictions in the error nlysis of numericl qudrture for both simple nd multiple integrls s well s for the Riemnn-Stieltjes integrl re given In Chpter, uthored by SS Drgomir nd TM Rssis, generlistions of the Ostrowski integrl inequlity for mppings of bounded vrition nd for bsolutely continuous functions vi kernels with n brnches including pplictions for generl qudrture formule, re given Chpter, uthored by A Sofo, builds on the work in Chpter He investigtes generlistions of integrl inequlities for n-times differentible mppings With the id of the modern theory of inequlities nd by use of generl Peno kernel, explicit bounds for interior point rules re obtined Firstly, he develops integrl equlities which re then used to obtin inequlities for n-times differentible mppings on the Lebesgue spces L, b, L p, b, < p < nd L, b Secondly, some prticulr inequlities re obtined which include explicit bounds for perturbed trpezoid, midpoint, Simpson s, Newton-Cotes, left nd right rectngle rules Finlly, inequlities re lso pplied to vrious composite qudrture rules nd the nlysis llows the determintion of the prtition required for the ccurcy of the result to be within prescribed error tolernce In Chpter 3, uthored by P Cerone nd SS Drgomir, unified tretment of three point qudrture rules is presented in which the clssicl rules of mid-point, trpezoidl nd Simpson type re recptured s prticulr cses Riemnn integrls re pproximted for the derivtive of the integrnd belonging to vriety of norms The Grüss inequlity nd number of vrints re lso presented which provide vriety of inequlities tht re suitble for numericl implementtion Mppings tht re of bounded totl vrition, Lipschitzin nd monotonic re lso investigted with reltion to Riemnn-Stieltjes integrls Explicit priori bounds re provided llowing the determintion of the prtition required to chieve prescribed error tolernce

6 CONTENTS vi It is demonstrted tht with the bove clsses of functions, the verge of midpoint nd trpezoidl type rule produces the best bounds In Chpter 4, uthored by P Cerone, product brnches of Peno kernels re used to obtin results suitble for numericl integrtion In prticulr, identities nd inequlities re obtined involving evlutions t n interior nd t the end points It is shown how previous work nd rules in numericl integrtion re recptured s prticulr instnces of the current development Explicit priori bounds re provided llowing the determintion of the prtition required for chieving prescribed error tolernce In the min, Ostrowski-Grüss type inequlities re used to obtin bounds on the rules in terms of vriety of norms In Chpter 5, uthored by NS Brnett, P Cerone nd SS Drgomir, new results for Ostrowski type inequlities for double nd multiple integrls nd their pplictions for cubture formule re presented This work is then continued in Chpter 6, uthored by G Hnn, where n Ostrowski type inequlity in two dimensions for double integrls on rectngle region is developed The resulting integrl inequlities re evluted for the clss of functions with bounded first derivtive They re employed to pproximte the double integrl by one dimensionl integrls nd function evlutions using different types of norms If the one-dimensionl integrls re not known, they themselves cn be pproximted by using suitble rule, to produce cubture rule consisting only of smpling points In ddition, some generlistions of n Ostrowski type inequlity in two dimensions for n - time differentible mppings re given The result is n integrl inequlity with bounded n - time derivtives This is employed to pproximte double integrls using one dimensionl integrls nd function evlutions t the boundry nd interior points In Chpter 7, uthored by John Roumeliotis, weighted qudrture rules re investigted The results re vlid for generl weight functions The robustness of the bounds is explored for specific weight functions nd for vriety of integrnds A comprison of the current development is mde with trditionl qudrture rules nd it is demonstrted tht the current development hs some dvntges In prticulr, this method llows the nodes nd weights of n n point rule to be esily obtined, which my be preferentil if the region of integrtion vries Other explicit error bounds my be obtined in dvnce, thus mking it possible to determine the weight dependent prtition required to chieve certin error tolernce In the lst chpter, SS Drgomir presents recent results in pproximting the Riemnn-Stieltjes integrl by the use of Trpezoid type, Ostrowski type nd Grüss type inequlities Applictions for certin clsses of weighted integrls re lso given This book is intended for use in the fields of integrl inequlities, pproximtion theory, pplied mthemtics, probbility theory nd sttistics nd numericl nlysis

7 vii The Editors, Melbourne nd Athens, December 000

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9 CHAPTER Generlistions of Ostrowski Inequlity nd Applictions by SS DRAGOMIR nd TM RASSIAS Abstrct Generliztions of Ostrowski integrl inequlity for mppings of bounded vrition nd for bsolutely continuous functions vi kernels with n brnches plus pplictions for generl qudrture formule re given Introduction The following result is known in the literture s Ostrowski s inequlity see for exmple, p 468) Theorem Let f :, b R be differentible mpping on, b) with the property tht f t) M for ll t, b) Then ) f x) b ) f t) dt b x +b 4 + b ) b ) M for ll x, b The constnt 4 is the best possible in the sense tht it cnnot be replced by smller constnt A simple proof of this fct cn be done by using the identity: ) f x) b where f t) dt + b t if t x p x, t) : t b if x < t b p x, t) f t) dt, x, b, which lso holds for bsolutely continuous functions f :, b R The following Ostrowski type result for bsolutely continuous functions holds see 7, 0 nd 8)

10 GENERALISATIONS OF OSTROWSKI INEQUALITY AND APPLICATIONS Theorem Let f :, b R be bsolutely continuous on, b Then, for ll x, b, we hve: f x) 3) f t) dt b 4 + x +b ) b b ) f if f L, b ; where r nd p+) p x b ) p+ + + x +b b f ; The constnts 4, Theorem b x b ) p+ p b ) p f q if f L q, b, r, ) re the usul Lebesgue norms on L r, b, ie, g : ess sup g t) t,b g r : g t) r dt p+) p nd ) r, r, ) p + q, p > ; respectively re shrp in the sense presented in The bove inequlities cn lso be obtined from the Fink result in on choosing n nd performing some pproprite computtions If one drops the condition of bsolute continuity nd ssumes tht f is Hölder continuous, then one my stte the result see 5) Theorem 3 Let f :, b R be of r H Hölder type, ie, 4) f x) f y) H x y r, for ll x, y, b, where r 0, nd H > 0 re fixed Then for ll x, b we hve the inequlity: f x) 5) f t) dt b b ) r+ ) r+ H x x + b ) r r + b b The constnt r+ is lso shrp in the bove sense Note tht if r, ie, f is Lipschitz continuous, then we get the following version of Ostrowski s inequlity for Lipschitzin functions with L insted of H) due to SS Drgomir 3, see lso ) 6) f x) b Here the constnt 4 is lso best f t) dt x +b 4 + b ) b ) L

11 3 SS Drgomir nd TM Rssis Moreover, if one drops the condition of the continuity of the function, nd ssumes tht it is of bounded vrition, then the following result due to Drgomir my be stted see lso 4 or ) Theorem 4 Assume tht f :, b R is of bounded vrition nd denote by b its totl vrition Then 7) f x) b for ll x, b The constnt is the best possible f t) dt + x +b b b f) If we ssume more bout f, ie, f is monotoniclly incresing, then the inequlity 7) my be improved in the following mnner see lso ) Theorem 5 Let f :, b R be monotonic nondecresing Then for ll x, b, we hve the inequlity: 8) f x) f t) dt b { x + b) f x) + b sgn t x) f t) dt {x ) f x) f ) + b x) f b) f x)} b + x +b f b) f ) b } All the inequlities in 8) re shrp nd the constnt is the best possible For recent generlistion of this result see 6 where further extensions were given The min im of the present chpter is to provide number of generlistions for kernels with N brnches of the bove Ostrowski type inequlity Nturl pplictions for qudrture formule re lso given Generlistions for Functions of Bounded Vrition Some Inequlities We strt with the following theorem 4 Theorem 6 Let I k : x 0 < x < < x k < x k b be division of the intervl, b nd α i i 0,, k + ) be k+ points so tht α 0, α i x i, x i i,, k) nd α k+ b If f :, b R is of bounded vrition on, b, then

12 GENERALISATIONS OF OSTROWSKI INEQUALITY AND APPLICATIONS 4 we hve the inequlity: 9) f x) dx ν h) + mx ν h) b f), k α i+ α i ) f x i ) { α i+ x } i + x i+ b, i 0,, k f) where ν h) : mx {h i i 0,, k }, h i : x i+ x i i 0,, k ) nd b f) is the totl vrition of f on the intervl, b Proof Define the kernel K :, b R given by see lso 4) t α, t, x ) t α, t x, x ) K t) : t α k, t x k, x k ) t α k, t x k, b Integrting by prts in Riemnn-Stieltjes integrl, we hve successively K t) df t) k xi+ k k K t) df t) x i t α i+ ) f t) xi+ x i xi+ x i xi+ x i t α i+ ) df t) f t) dt k α i+ x i ) f x i ) + x i+ α i+ ) f x i+ ) k α ) f ) + α i+ x i ) f x i ) k + i x i+ α i+ ) f x i+ ) + b α n ) f b) k k α ) f ) + α i+ x i ) f x i ) + x i α i ) f x i ) + b α n ) f b) i f t) dt k α ) f ) + α i+ α i ) f x i ) + b α n ) f b) i k α i+ α i ) f x i+ ) f t) dt, i f t) dt f t) dt f t) dt

13 5 SS Drgomir nd TM Rssis nd then we hve the integrl equlity which is of interest in itself too: 0) f t) dt k α i+ α i ) f x i ) Using the modulus properties, we hve k K t) df t) k However, xi+ t α i+ ) f t) dt sup x i t x i,x i+ xi+ x i xi+ x i K t) df t) K t) df t) K t) df t) : T x i+ t α i+ f) x i+ mx {α i+ x i, x i+ α i+ } f) x i Then T x i+ x i ) + α i+ x i + x i+ x i x i+ f)dt k h i + α i+ x x i + x i+ i+ f) x i mx,,k h i + α i+ x i + x i+ k x i+ f) x i { ν h) + mx α i+ x } i + x i+ b, i 0,, k f) : V x i Now, s then mx nd, consequently, α i+ x i + x i+ h i, { α i+ x i + x i+ The theorem is completely proved V ν h) }, i 0,, k ν h) b f) Now, if we ssume tht the points of the division I k re given, then the best inequlity we cn obtin from Theorem 6 is embodied in the following corollry:

14 GENERALISATIONS OF OSTROWSKI INEQUALITY AND APPLICATIONS 6 Corollry 7 Let f nd I k be s bove Then we hve the inequlity: f x) dx ) x ) f ) k + x i+ x i ) f x i ) + b x k ) f b) i b ν h) f) Proof We choose in Theorem 6, In this cse we get α 0, α + x, α x + x,, α k x k + x k, α k x k + x k k α i+ α i ) f x i ) α α 0 ) f ) + α α ) f x ) + nd α k+ b + α k α k f x k ) + b α k ) f b) ) + x x + x f ) + + x ) f x ) xk + b + + x ) k + x k f x k ) + b x ) k + b f b) k x ) f ) + x i+ x i ) f x i ) + b x k ) f b) i Now, pplying the inequlity 9), we get ) The following corollry for equidistnt prtitioning lso holds Corollry 8 Let I k : x i : + b ) i i 0,, k) k be n equidistnt prtitioning of, b If f is s bove, then we hve the inequlity: b k f ) + f b) b ) k i) + ib ) f x) dx b ) + f k k k b k b ) f) i

15 7 SS Drgomir nd TM Rssis A Generl Qudrture Formul Let n : x n) 0 < x n) < < x n) n < xn) n b be sequence of division of, b nd consider the sequence of numericl integrtion formule I n f, n, w n ) : n j0 w n) j f x n) j where w n) j j 0,, n) re the qudrture weights nd n j0 wn) j b The following theorem provides sufficient condition for the weights w n) j I n f, n, w n ) pproximtes the integrl f x) dx see lso 4) ) so tht Theorem 9 Let f :, b R be function of bounded vrition on, b If the qudrture weights w n) j stisfy the condition i 3) x n) i w n) j x n) i+ for ll i 0,, n, j0 then we hve the estimte I 4) n f, n, w n ) f x) dx h ν n)) i + mx + ν h n)) b f), j0 w n) j xn) i + x n) b i+, i 0,, n f) where ν h n)) { } : mx h n) i i 0,, n nd h n) i : x n) i+ xn) i In prticulr, 5) lim I n f, n, w n ) νh n) ) 0 uniformly by rpport of the w n Proof Define the sequence of rel numbers Note tht α n) α n) nd observe lso tht α n) i+ i+ : + i n+ + n x n) i j0 j0, x n) i+ f x) dx w n) j, i 0,, n w n) j + b b, Define α n) 0 : nd compute α n) α n) 0,

16 Then GENERALISATIONS OF OSTROWSKI INEQUALITY AND APPLICATIONS 8 α n) i+ αn) i + n i j0 α n) n+ αn) n ) α n) i+ αn) i f w n) j i j0 w n) j n b + x n) i ) n j0 w n) i i,, n ), w n) j w n) i f x n) i Applying the inequlity 9), we get the estimte 4) w n) n ) I n f, n, w n ) The uniform convergence by rpport of qudrture weights w n) j lst inequlity is obvious by the Now, consider the equidistnt prtitioning of, b given by E n : x n) i : + i b ) i 0,, n) n nd define the sequence of numericl qudrture formule n I n f, w n ) : w n) i f + in b ) The following corollry which cn be more useful in prctice holds: Corollry 0 Let f be s bove If the qudrture weight w n) j condition: 6) i n b i j0 then we hve: I 7) n f, w n ) f x) dx b n + mx i + b ) n b f) j0 In prticulr, we hve the limit uniformly by rpport of w n w n) j i +, i 0,, n ; n w n) j i + lim I n f, w n ) n b ) n f x) dx, stisfy the b, i 0,, n f)

17 9 SS Drgomir nd TM Rssis 3 Prticulr Inequlities The following proposition holds 4 see lso 4) Proposition Let f :, b R be function of bounded vrition on, b Then we hve the inequlity: 8) f x) dx α ) f ) + b α) f b) b ) + α + b b f) for ll α, b The proof follows by Theorem 6 choosing x 0, x b, α 0, α α, b nd α b Remrk ) If in 8) we put α b, then we get the left rectngle inequlity 9) b f x) dx b ) f ) b ) f); b) If α, then by 8) we get the right rectngle inequlity b 0) f x) dx b ) f b) b ) f); c) It is esy to see tht the best inequlity we cn get from 8) is for α +b obtining the trpezoid inequlity see lso 4) ) f x) dx f ) + f b) b ) b b ) f) Another proposition with mny interesting prticulr cses is the following one 4 see lso 4): Proposition Let f be s bove nd x b, α x α b Then we hve ) f x) dx α ) f ) + α α ) f x ) + b α ) f b) b ) + x + b + α + x + α x + b + α + x α x + b b f) b ) + x + b b b f) b ) f)

18 GENERALISATIONS OF OSTROWSKI INEQUALITY AND APPLICATIONS 0 Proof Consider the division x 0 x x b nd the numbers α 0, α, x, α x, b nd α 3 b Now, pplying Theorem 6, we get f x) dx α ) f ) + α α ) f x ) + b α ) f b) { mx {x, b x } + mx α + x, α x } + b b f) 4 b ) + x + b + α + x + α x + b + α + x α x + b b f) nd the first inequlity in ) is proved Now, let observe tht α + x x, α x + b b x Consequently, mx { α + x nd the second inequlity in ) is proved The lst inequlity is obvious, α x } + b mx {x, b x } Remrk ) If we choose bove α, α b, then we get the following Ostrowski type inequlity obtined by Drgomir in the recent pper : 3) f x) dx b ) f x ) b ) + x + b b f) for ll x, b We note tht the best inequlity we cn get in 3) is for x +b obtining the midpoint inequlity see lso ) ) + b 4) f x) dx f b ) b b ) f) b) If we choose in ) α 5+b 6, α +5b 6 nd x 5+b 6, +5b 6, then we get f x) dx b f ) + f b) + f x ) 5) 3 b ) + x + b { + mx x + b } 3, + b b x f) 3

19 SS Drgomir nd TM Rssis In prticulr, if we choose in 5), x +b, then we get the following Simpson s inequlity 9 f x) dx b ) f ) + f b) + b 6) + f 3 b 3 b ) f) 4 Prticulr Qudrture Formule Let us consider the prtitioning of the intervl, b given by n : x 0 < x < < x n < x n b nd put h i : x i+ x i i 0,, n ) nd ν h) : mx {h i i 0,, n } The following theorem holds 4: Theorem 3 Let f :, b R be bsolutely continuous on, b nd k Then we hve the composite qudrture formul 7) where 8) A k n, f) : T n, f) + k nd f x) dx A k n, f) + R k n, f), n k k j) xi + jx i+ f k j 9) T n, f) : n f x i ) + f x i+ ) h i is the trpezoid qudrture formul The reminder R k n, f) stisfies the estimte 30) R k n, f) b k ν h) f) h i get Proof Applying Corollry 8 on the intervls x i, x i+ i 0,, n ) we xi+ x i x k h i+ i f) f x) dx k x i f x i ) + f x i+ ) h i + h i k k k j) xi + jx i+ f k j

20 GENERALISATIONS OF OSTROWSKI INEQUALITY AND APPLICATIONS Now, using the generlized tringle inequlity, we get: R k n, f) n xi+ f x) dx x i k f x i) + f x i+ ) h i + h k i k j) xi + jx i+ f k k n k f) ν h) n k h i x i+ x i nd the theorem is proved x i+ x i f) ν h) k b f) j The following corollries hold: Corollry 4 Let f be s bove Then we hve the formul: 3) f x) dx T n n, f) + M n n, f) + R n, f) where M n n, f) is the midpoint qudrture formul, n ) xi + x i+ M n n, f) : f h i nd the reminder R n, f) stisfies the inequlity: 3) R n, f) b 4 ν h) f) Corollry 5 Under the bove ssumptions we hve b f x) dx n ) xi + x i+ 33) T n n, f) + f h i 3 3 n ) xi + x i+ + f h i + R 3 n, f) 3 The reminder R 3 n, f) stisfies the bound: 34) R 3 n, f) b 6 ν h) f) The following theorem holds 4 see lso 4): Theorem 6 Let f nd n be s bove nd ξ i x i, x i+ i 0,, n ) Then we hve the qudrture formul: 35) n f x) dx ξ i x i ) f x i ) + x i+ ξ i ) f x i+ ) + R ξ, n, f)

21 3 SS Drgomir nd TM Rssis The reminder R ξ, n, f) stisfies the estimtion: 36) R ξ, n, f) { ν h) + mx ξ i x } i + x i+ b, i 0,, n f) ν h) for ll ξ i s bove b f), get Proof Apply Proposition on the intervl x i, x i+ i 0,, n ) to xi+ f x) dx ξ i x i ) f x i ) + x i+ ξ i ) f x i+ ) x i { h i + mx ξ i x } x i + x i+ i+ f) x i Summing over i from 0 to n, using the generlized tringle inequlity nd the properties of the mximum mpping, we get 36) Corollry 7 Let f nd n be s bove Then we hve ) the left rectngle rule 37) n f x) dx f x i ) h i + R l n, f) ; ) the right rectngle rule 38) n f x) dx f x i+ ) h i + R r n, f) ; 3) the trpezoid rule 39) where nd f x) dx T n, f) + R T n, f) b R l n, f) R r n, f) ν h) f) R T n, f) b ν h) f) The following theorem lso holds 4

22 GENERALISATIONS OF OSTROWSKI INEQUALITY AND APPLICATIONS 4 Theorem 8 Let f nd n be s bove nd ξ i x i, x i+, x i α ) i x i+, then we hve the qudrture formul: α ) i 40) n f x) dx n + α ) i x i ) f x i ) + x i+ α ) i n ) f x i+ ) + R ) α ) i α ) i f ξ i ) ) ξ, α ), α ), n, f ξ i The reminder R ξ, α ), α ), n, f ) stisfies the estimtion R ξ, α ), α ) 4), n, f) { ν h) + mx,,n ξ i x i + x i+ { + mx mx,,n α) i x i + ξ i, mx,,n α) i ξ }} i + x i+ b f) ν h) + mx,n ξ i x i + x i+ b f ν h) f) Proof Apply Proposition on the intervl x i, x i+ to obtin xi+ f x) dx x i ) ) ) α ) i x i f x i ) + α ) i α ) i f ξ i ) + x i+ α ) i f x i+ ) h i + ξ i x i + x i+ { + mx α ) i x i + ξ i, α ) i ξ } x i + x i+ i+ f) x i Summing over i from 0 to n nd using the properties of modulus nd mximum, we get the desired inequlity We shll omit the detils The following corollry is the result of Drgomir from the recent pper Corollry 9 Under the bove ssumptions, we hve the Riemnn s qudrture formul: 4) n f x) dx f ξ i ) h i + R R ξ, n, f)

23 5 SS Drgomir nd TM Rssis The reminder R R ξ, n, f) stisfies the bound 43) R R ξ, n, f) { ν h) + mx ξ i x } i + x i+ b, i 0,, n f) ν h) b f) for ll ξ i x i, x i+ i 0,, n) Finlly, the following corollry which generlizes Simpson s qudrture formul holds Corollry 0 Under the bove ssumptions nd if ξ i xi++5x i 6, xi+5xi+ 6 i 0,, n ), then we hve the formul: 44) f x) dx n f x i ) + f x i+ ) h i + n f ξ 6 3 i ) h i + S f, n, ξ) The reminder S f, n, ξ) stisfies the estimte: 45) S f, n, ξ) { { ν h) + mx ξ i x } i + x i+,,n { + mx mx,,n ξ i x i + x i+ 3, mx,,n x i + x i+ 3 The proof follows by the inequlity 5) nd we omit the detils ξ i }} b Remrk 3 Now, if we choose in 44), ξ i xi+xi+, then we get Simpson s qudrture formul 9 46) f x) dx n f x i ) + f x i+ ) h i 6 + n ) xi + x i+ f h i + S f, n ) 3 where the reminder term S f, n ) stisfies the bound: 47) S f, n ) b 3 ν h) f) f) 3 Generlistions for Functions whose Derivtives re in L 3 Some Inequlities We strt with the following result 5 Theorem Let I k : x 0 < x < < x k < x k b be division of the intervl, b, α i i 0,, k + ) be k + points so tht α 0, α i

24 GENERALISATIONS OF OSTROWSKI INEQUALITY AND APPLICATIONS 6 x i, x i i,, k) nd α k+ b If f :, b R is bsolutely continuous on, b, then we hve the inequlity: 48) k f x) dx α i+ α i ) f x i ) k k h i + α i+ x ) i + x i+ f 4 k f h i b ) f ν h), where h i : x i+ x i i 0,, k ) nd ν h) : mx {h i i 0,, k } The constnt 4 in the first inequlity nd the constnt in the second nd third inequlity re the best possible Proof Define the mpping K :, b R given by see the proof of Theorem 6) K t) : t α, t, x ) ; t α, t x, x ) ; t α k, t x k, x k ) ; t α k, t x k, b Integrting by prts, we hve successively: K t) f t) dt k xi+ k k K t) f t) dt x i t α i+ ) f t) xi+ x i xi+ x i xi+ x i t α i+ ) f t) dt f t) dt k α i+ x i ) f x i ) + x i+ α i+ ) f x i+ ) f t) dt k k α ) f ) + α i+ x i ) f x i ) + x i+ α i+ ) f x i+ ) + b α n ) f b) i f t) dt

25 7 SS Drgomir nd TM Rssis k k α ) f ) + α i+ x i ) f x i ) + x i α i ) f x i ) + b α n ) f b) i f t) dt k α ) f ) + α i+ α i ) f x i ) + b α n ) f b) i k α i+ α i ) f x i ) f t) dt nd then we hve the integrl equlity: k 49) f t) dt α i+ α i ) f x i ) Using the properties of modulus, we hve K t) f 50) t) dt k xi+ k K t) f t) dt x i k xi+ x i A simple clcultion shows tht 5) xi+ x i t α i+ dt for ll i 0,, k xi+ x i t α i+ f t) dt f αi+ i K t) f t) dt K t) f t) dt k xi+ x i xi+ f t) dt t α i+ dt α i+ t) dt + t α i+ ) dt x i α i+ x i+ α i+ ) + α i+ x i ) 4 h i + α i+ x ) i + x i+ Now, by 49) 5), we get the first inequlity in 48) Assume tht the first inequlity in 48) holds for constnt c > 0, ie, k 5) f x) dx α i+ α i ) f x i ) k k c h i + α i+ x ) i + x i+ f If we choose f :, b R, f x) x, α 0, α b, x 0, x b in 5), we obtin b ) c b ) b ) +, 4

26 GENERALISATIONS OF OSTROWSKI INEQUALITY AND APPLICATIONS 8 from where we get c 4, nd the shrpness of the constnt 4 is proved The lst two inequlities s well s the shrpness of the constnt we omit the detils re obvious nd Now, if we ssume tht the points of the division I k re given, then the best inequlity we cn get from Theorem is embodied in the following corollry: Corollry Let f, I k be s bove Then we hve the inequlity 53) f x) dx k x ) f ) + x i+ x i ) f x i ) + b x k ) f b) k 4 f h i i The constnt 4 is the best possible one Proof Similr to the proof of Corollry 7 The cse of equidistnt prtitioning is importnt in prctice Corollry 3 Let I k : x i +i b k i 0,, k) be n equidistnt prtitioning of, b If f is s bove, then we hve the inequlity b k f ) + f b) b ) k i) + ib 54) f x) dx b ) + f k k k 4k b ) f The constnt 4 is the best possible one Remrk 4 If k, then we hve the inequlity see for exmple 4) f ) + f b) 55) f x) dx b ) 4 b ) f Choose f :, b R, f x) x +b, which is L-Lipschitzin with L nd if t ), +b f x) if t +b, b Then f nd f x) dx f ) + f b) b ) i b ), 4 nd the equlity is obtined in 55), showing tht the constnt 4 is shrp

27 9 SS Drgomir nd TM Rssis 3 A Generl Qudrture Formul Let n : x n) 0 < x n) < < x n) n < xn) n b be sequence of divisions of, b nd consider the sequence of numericl integrtion formule where w n) j b I n f, n, w n ) : n j0 w n) j f x n) j j 0,, n) re the qudrture weights nd ssume tht n j0 wn) j The following theorem contins sufficient condition for the weights w n) j so tht I n f, n, w n ) pproximtes the integrl f x) dx with n error expressed in terms of f see lso 5) Theorem 4 Let f :, b R be n bsolutely continuous mpping on, b If the qudrture weights w n) j j 0,, n) stisfy the condition i 56) x n) i w n) j x n) i+ for ll i 0,, n ; j0 then we hve the estimtion I 57) n f, n, w n ) f x) dx n n i h n) i f n h n) i j0 j0 w n) j xn) i ), f b ) ν h n)), where ν h n)) : mx{h n) i : i 0,, n } nd h n) i In prticulr, if f <, then lim I n f, n, w n ) νh n) ) 0 uniformly by the influence of the weights w n + x n) i+ f : x n) i+ xn) i f x) dx Proof Similr to the proof of Theorem 9 nd we omit the detils The cse when the prtitioning is equidistnt is importnt in prctice Consider, then, the prtitioning E n : x n) i : + i b i 0,, n), n nd define the sequence of numericl qudrture formule n I n f, w n ) : w n) i f + i b ) n, w n) j b n j0

28 GENERALISATIONS OF OSTROWSKI INEQUALITY AND APPLICATIONS 0 The following result holds: Corollry 5 Let f :, b R be bsolutely continuous on, b If the qudrture weights w n) i stisfy the condition: i n i w n) j i + i 0,, n ); b n j0 then we hve the estimte I n f, w n ) f x) dx f n 4n b i ) + n f b ) In prticulr, if f <, then uniformly by the influence of w n lim I n f, w n ) n j0 w n) j i + f x) dx b n 33 Prticulr Inequlities In this sub-section we point out some prticulr inequlities which generlize some clssicl results such s: rectngle inequlity, trpezoid inequlity, Ostrowski s inequlity, midpoint inequlity, Simpson s inequlity nd others in terms of the sup-norm of the derivtive 5 see lso 4) Proposition 6 Let f :, b R be bsolutely continuous on, b nd α, b Then we hve the inequlity: 58) f x) dx α )f ) + b α)f b) 4 b ) + α + b ) f b ) f The constnt 4 is the best possible one Proof Follows from Theorem by choosing x 0, x b, α 0, α α, b nd α b Remrk 5 ) If in 58) we put α b, then we get the left rectngle inequlity : 59) f x) dx b ) f ) b ) f

29 SS Drgomir nd TM Rssis b) If α, then by 58) we obtin the right rectngle inequlity 60) f x) dx b ) f ) b ) f 6) c) It is cler tht the best estimtion we cn hve in 58) is for α +b getting the trpezoid inequlity see lso ) f x) dx f ) + f b) b ) 4 b ) f Another prticulr integrl inequlity with mny pplictions is the following one 5: Proposition 7 Let f :, b R be n rbitrry bsolutely continuous mpping on, b nd α x α b Then we hve the inequlity: 6) f x) dx α )f ) + α α )fx ) + b α ) f b) 8 b ) + x + b ) + α + x ) + α x ) + b f Proof Follows by Theorem nd we omit the detils Corollry 8 Let f be s bove nd x, b Then we hve Ostrowski s inequlity: 63) f x) dx b ) fx ) 4 b ) + x + b ) f Remrk 6 If we choose x +b in 63), we obtin the midpoint inequlity 64) ) + b f x) dx b ) f 4 b ) f The following corollry generlizing Simpson s inequlity holds: Corollry 9 Let f be s bove nd x 5+b 6, +5b 6 Then we hve the inequlity f x) dx b f ) + f b) + fx ) 65) b ) + x + b ) f Proof Follows by Proposition 7 using simple computtion nd we omit the detils

30 GENERALISATIONS OF OSTROWSKI INEQUALITY AND APPLICATIONS Remrk 7 Let us observe tht the best estimtion we cn obtin from 65) is tht one for which x +b, obtining the Simpson s inequlity 0 66) f x) dx b 3 f ) + f b) + b + f ) 5 36 b ) f The following corollry lso holds Corollry 30 Let f be s bove nd α +b α b Then we hve the inequlity 67) 8 b ) + f x) dx α )f ) + α α )f α 3 + b ) + α + 3b b ) ) + b α )f b) f The proof is obvious by Proposition 7 by choosing x +b Remrk 8 The best estimtion we cn obtin from 67) is tht one for which α 3+b 4 nd α +3b 4, obtining the inequlity 3 68) f x) dx b f ) + f b) + b + f ) 8 b ) f The following proposition generlizes the three-eights rule of Newton-Cotes: Proposition 3 Let f be s bove nd x x b nd α, x, α x, x, α 3 x, b Then we hve the inequlity 69) f x) dx α )f ) + α α )fx ) +α 3 α )fx ) + b α 3 )f b) 4 x ) + x x ) + b x ) + α + x ) + α x ) + x + α 3 x ) + b f The proof is obvious by Theorem The next corollry contins generliztion of the three-eights rule of Newton- Cotes in the following wy:

31 3 SS Drgomir nd TM Rssis Corollry 3 Let f be s bove nd α +b 3 α b+ 3 α 3 b Then we hve the inequlity: ) + b 70) f x) dx α )f ) + α α )f 3 ) + b +α 3 α )f + b α 3 )f b) 3 b ) + α 5 + b ) 6 + α + b ) + α 3 + 5b ) f 6 The proof follows by the bove proposition by choosing x +b 3 nd x +b 3 Remrk 9 ) ) Now, if we choose α b+7 8, α +b nd α 3 +7b 8 in 70), then we get the three-eights rule of Newton-Cotes 7) f x) dx b b ) f f ) + 3f + b 3 ) + 3f + b 3 ) + f b) b) The best estimtion we cn get from 70) is tht one for which α 5+b 6, α +b, α 3 +5b 6 obtining the inequlity f x) dx b ) ) + b + b f ) + f + f + f b) 7) b ) f 34 Prticulr Qudrture Formule Let us consider the prtitioning of the intervl, b given by n : x 0 < x < < x n < x n b nd put h i : x i+ x i i 0,, n ) nd νh) : mx{h i i 0,, n } The following theorem holds 5: Theorem 33 Let f :, b R be bsolutely continuous on, b nd k Then we hve the composite qudrture formul 73) f x) dx A k n, f) + R k n, f), where 74) A k n, f) : n k k j)xi + jx i+ T n, f) + f k k j h i

32 GENERALISATIONS OF OSTROWSKI INEQUALITY AND APPLICATIONS 4 nd 75) T n, f) : n f x i ) + f x i+ ) h i is the trpezoid qudrture formul The reminder R k n, f) stisfies the estimte 76) R k n, f) n 4k f h i Proof Applying Corollry on the intervls x i, x i+ i 0,, n ), we obtin xi+ f x) dx x i k f x i) + f x i+ ) h i + h k i k j)xi + jx i+ f k k 4k h i f Summing over i from 0 to n nd using the generlized tringle inequlity, we get the desired estimtion 76) The following corollry holds: Corollry 34 Let f, n be s bove Then we hve the qudrture formul j f x) dx T n, f) + M n, f) + R n, f), where M n, f) is the midpoint rule: n ) xi + x i+ M n, f) : f h i The reminder stisfies the estimte: 77) R n, f) n 8 f h i The following corollry lso holds: Corollry 35 Let f, n be s bove Then we hve the qudrture formul b f x) dx n ) xi + x i+ 78) T n, f) + f h i 3 n ) xi + x i+ + f h i + R 3 n, f), 3 where the reminder R 3 n, f) stisfies the bound: 79) R n, f) n f h i

33 5 SS Drgomir nd TM Rssis The following theorem holds s well 5 see lso 4): Theorem 36 Let f nd n be s bove Suppose tht ξ i x i, x i+ i 0,, n ) Then we hve the formul 80) n f x) dx ξ i x i ) f x i ) + x i+ ξ i ) f x i+ ) + Rξ, n, f) The reminder Rξ n, n, f) stisfies the estimte: n n Rξ, n, f) h i + ξ 4 i x ) i + x i+ 8) f get n f h i Proof Apply Proposition 6 on the intervls x i, x i+ i 0,, n ) to xi+ f x) dx ξ i x i ) f x i ) + x i+ ξ i )f x i+ ) x i 4 h i + ξ i x ) i + x i+ f f h i Summing over i from 0 to n nd using the generlized tringle inequlity we deduce the desired inequlity 8) Corollry 37 8) 83) 84) i) The left rectngle rule : Let f nd n be s bove Then we hve ii) The right rectngle rule : iii) The trpezoid rule : where nd n f x) dx f x i ) h i + R l n, f) ; n f x) dx f x i+ ) h i + R r n, f) ; The following theorem lso holds 5 f x) dx T n, f) + R T n, f) R l n, f), R r n, f) n f R T n, f) n 4 f h i h i

34 GENERALISATIONS OF OSTROWSKI INEQUALITY AND APPLICATIONS 6 Theorem 38 Let f nd n be s bove If x i α ) i ξ i α ) i x i+ i 0,, n ), then we hve the formul: 85) n f x) dx n + α ) i x i ) f x i ) + x i+ α ) i where the reminder stisfies the bound R ξ, α ), α ) 86), n, f) n 8 n + n ) f x i+ ) + R n ξ i x i + x i+ h i + α ) i x i + ξ i ) n + ) α ) i α ) i f ξ i ) ) ) ξ, α ), α ), n, f, α ) i ξ ) i + x i+ f The proof follows by Proposition 7 pplied for the intervls x i, x i+ i 0,, n ) nd we omit the detils The following corollry of the bove theorem holds 0 Corollry 39 Let f, n be s bove nd ξ i x i, x i+ i 0,, n ) Then we hve the formul of Riemnn s type: 87) n f x) dx f ξ i ) h i + R R ξ, n, f) where the reminder, R R ξ, n, f), stisfies the estimte n n R R ξ, n, f) h i + ξ 4 i x ) i + x i+ 88) f n f h i Remrk 0 If we choose in 87), ξ i qudrture formul 0 where f x) dx M n, f) + R M n, f), R M n, f) n 4 f h i xi+xi+, then we get the midpoint The following corollry holds s well 5 see lso 3)

35 7 SS Drgomir nd TM Rssis Corollry 40 Let f, n be s bove nd ξ i Then we hve the formul 89) f x) dx n f x i ) + f x i+ ) h i 6 5xi+x i+ 6, xi+5xi+ 6 i 0,, n ) + n f ξ 3 i ) h i + R s ξ, n, f), where the reminder, R s ξ, n, f), stisfies the inequlity: n 5 n 90) R s ξ, n, f) h i + ξ 36 i x ) i + x i+ f Remrk If we choose bove ξ i xi+xi+ i 0,, n ), then we obtin 9) f x) dx n f x i ) + f x i+ ) h i 6 + n ) xi + x i+ f h i + R S n, f), 3 where the reminder stisfies the bound 0 9) R S n, f) 5 n 36 f h i The following corollry holds too Corollry 4 Let f, n be s bove nd x i α ) i i 0,, n ) Then we hve the formul 93) n f x) dx n + α ) i x i ) f x i ) + x i α ) i n ) α ) i α ) i f ) f x i+ ) + R B α ), α ), n, f where the reminder stisfies the estimte: ) n R B α ) n, α ) n, n, f h i + 8 n xi+xi+ α ) i x i+ ) xi + x i+ ), α ) i 3x i + x i+ 4 n + α ) i x i + 3x i+ 4 ) ) f Finlly, the following theorem holds 5:

36 GENERALISATIONS OF OSTROWSKI INEQUALITY AND APPLICATIONS 8 Theorem 4 Let f, n be s bove nd x i ξ ) i ξ ) i x i+ nd α ) i x i, ξ ) i, α ) i ξ ) i, ξ ) i nd α 3) i ξ ) i, x i+ for i 0,, n Then we hve formul: 94) n n + +R f x) dx α ) i x i ) f x i ) + ) α 3) i α ) i f n ξ ) i ) α ) i α ) i f ) + n ) ξ ), ξ ), α ), α ), α 3), n, f ξ ) i x i+ α 3) i nd the reminder stisfies the estimtion R ξ ), ξ ), α ), α ), α 3), n, f) n ) n ) n ξ ) i x i + ξ ) i ξ ) i + 4 n + n + ) i x i+ + ξ ) n i + ) ξ) i + x i+ f α ) α 3) i α ) i The proof follows by Proposition 3 We omit the detils ) ξ) i + ξ ) i ) f x i+ ) x i+ ξ ) i Remrk We note only tht if we choose α ) i xi++7xi 8, α ) i xi+xi+, α 3) i xi+7xi+ 8, ξ ) i xi+xi+ 3 nd ξ ) i xi+xi+ 3 i 0,, n ) then we get the three-eights formul of Newton-Cotes: f x) dx 8 n f x i ) + 3f +R N C n, f), where the reminder stisfies the bound xi + x i+ 3 ) + 3f R N C n, f) 5 n 88 f h i ) xi + x i+ 3 ) ) + f x i+ ) 4 Generlistion for Functions whose Derivtives re in L p 4 Some Inequlities We strt with the following result 7, Theorem 43 Let I k : x 0 < x < < x k < x k b be division of the intervl, b nd α i i 0,, k + ) be k + points such tht α 0,

37 9 SS Drgomir nd TM Rssis α i x i, x i i,, k) nd α k+ b If f :, b R is bsolutely continuous on, b, then we hve the inequlity: 95) f x) dx q + ) q q + ) q k α i+ α i ) f x i ) k f p α i+ x i ) q+ + x i+ α i+ ) q+ q k f p h q+ i q ν h) b ) q q + ) q f p where h i : x i+ x i i 0,, k ), ν h) : mx {h i i 0,, n}, p >, p + q, nd p is the usul L p, b norm Proof Consider the kernel K :, b R given by see lso Theorem ): K t) : Integrting by prts, we hve the identity: t α, t, x ); t α, t x, x ); t α k, t x k, x k ); t α k, t x k, b 96) f t) dt k α i+ α i ) f x i ) K t) f t) dt On the other hnd, we hve 97) K t) f t) dt k xi+ x i k xi+ x i k xi+ x i K t) f t) dt K t) f t) dt t α i+ f t) dt Using Hölder s integrl inequlity for p >, p + q, we cn write 98) xi+ xi+ ) t α i+ f t) dt t α i+ q q x i+ ) dt f t) p q dt x i x i x i

38 GENERALISATIONS OF OSTROWSKI INEQUALITY AND APPLICATIONS 30 However, xi+ x i αi+ t α i+ q dt xi+ α i+ t) q dt + t α i+ ) q dt x i α i+ α i+ x i ) q+ + x i+ α i+ ) q+ q + ) nd then, by the inequlity 98), we deduce 99) xi+ x i q + ) q t α i+ f t) dt α i+ x i ) q+ + x i+ α i+ ) q+ xi+ q x i ) f t) p q dt By reltions 97)-99) nd by Hölder s discrete inequlity, we obtin K t) f t) dt k α q + ) i+ x i ) q+ + x i+ α i+ ) q+ xi+ ) q f t) p p dt q x i k q + ) q k xi+ q + ) q α i+ x i ) q+ + x i+ α q+ ) q q q i+) x i k ) ) p p f t) p p dt nd the first inequlity in 95) is proved α i+ x i ) q+ + x i+ α i+ ) q+) q f p Now, consider the function g : α, β R, g t) t α) q+ + β t) q+ Then g t) q + ) t α) q β t) q, g t) 0 iff t α+β nd g t) < 0 if t α, α+β ) nd g t) > 0 if t α+β, β, which shows tht ) α + β 00) inf g t) g β α)q+ t α,β q nd 0) sup g t) g α) g β) β α) q+ t α,β

39 3 SS Drgomir nd TM Rssis Using the bove bound 0), we my write k α i+ x i ) q+ + x i+ α i+ ) q+ k nd the second inequlity in 95) is obtined For the lst inequlity we only remrk tht k h q+ i The theorem is completely proved k ν q h) h i b ) ν q h) h q+ i, Now, if we ssume tht the points of the division I k re fixed, then the best inequlity we cn obtin from Theorem 43 is embodied in the following corollry Corollry 44 Let I k : x 0 < x < < x k < x k b be division of the intervl, b If f is s bove, then we hve the inequlity: 0) f x) dx k x ) f ) + x i+ x i ) f x i ) + b x k ) f b) q + ) q k f p i h q+ i q ν h) b ) q q + ) q The cse of equidistnt prtitioning is importnt in prctice Corollry 45 Let f p I k : x i : + i b i 0,, k), k be n equidistnt prtitioning of, b If f is s bove, then we hve the inequlity: b k f ) + f b) b ) k i) + ib 03) f x) dx b ) + f k k k b )+ q k q + ) q f p i 4 Generl Qudrture Formule Let n : x n) 0 < x n) < < x n) n < xn) n b be sequence of division of, b nd consider the sequence of numericl integrtion formule see Subsection 4) n 04) I n f, n, w n ) : j0 w n) j f x n) j where w n) j j 0,, n) re the qudrture weights nd n j0 wn) j b ),

40 GENERALISATIONS OF OSTROWSKI INEQUALITY AND APPLICATIONS 3 The following theorem contins sufficient condition for the weights w n) j such tht I n f, n, w n ) pproximtes the integrl f x) dx with n error expressed in terms of f p, p, ), 7 Theorem 46 Let f :, b R be n bsolutely continuous mpping on, b If the qudrture weights w n) j stisfy the condition 05) x n) i then we hve the estimte 06) I n f, n, w n ) f q + ) p q n + q + ) q i j0 i j0 w n) j f x) dx w n) j n f p h n) i x n) i+ for ll i 0,, n, x n) i q+ q q+ where ν h n)) { } : mx h n) i i 0,, n In prticulr, if f p <, then + lim I n f, n, w n ) νh n) ) 0 uniformly by rpport of the weight w n x n) i+ i ν h n)) b ) q q + ) q nd h n) i j0 w n) j f p, : x n) i+ xn) i f x) dx q+ /q Proof Similr to the proof of Theorem 9 nd we omit the detils The cse when the prtitioning is equidistnt is importnt in prctice Consider, then, the prtitioning E n : x n) i : + i b i 0,, n) nd define the sequence of numericl qudrture formule n I n f, w n ) : w n) i f + i n n b ) w n) j b j0 The following result holds:

41 33 SS Drgomir nd TM Rssis Corollry 47 Let f :, b R be bsolutely continuous on, b If the qudrture weights w n) j stisfy the condition: i i 07) n w n) j i +, i 0,, n ; n j0 then we hve the estimte: I 08) n f, w n ) f x) dx j0 f q + ) p q n i w n) j i b ) n b )+ q n q + ) q f p In prticulr, if f p <, then q+ 09) lim n I n f, w n ) uniformly by the influence of the weights w n + i + i n b ) f x) dx j0 w n) j q+ 43 Prticulr Inequlities In this sub-section we point out prticulr inequlities which generlize some clssicl results such s: Rectngle Inequlity, Trpezoid Inequlity, Ostrowski s Inequlity, Midpoint Inequlity, Simpson s Inequlity nd others in terms of the p norm of the derivtive 7 Proposition 48 Let f :, b R be bsolutely continuous on, b nd α, b Then we hve the inequlity see lso 4): 0) f x) dx α ) f ) + b α) f b) f q + ) p α ) q+ + b α) q+ q q b )+ q q + ) q f p Proof Follows from Theorem 43 by choosing x 0, x b, α 0, α α, b nd α b Remrk 3 ) ) If in 0) we put α b, then we obtin the left rectngle inequlity ) f x) dx b ) f ) b )+ q f q + ) p q q

42 GENERALISATIONS OF OSTROWSKI INEQUALITY AND APPLICATIONS 34 b) If α, then by 0) we hve the right rectngle inequlity ) f x) dx b ) f b) b )+ q f q + ) p q 3) c) It is cler tht the best estimte we cn hve in 0) is for α +b getting the trpezoid inequlity see lso 4): f x) dx f ) + f b) b ) b ) + q f q + ) p q Another prticulr integrl inequlity with mny pplictions is the following one see lso 3): Proposition 49 Let f :, b R be n bsolutely continuous mpping on, b nd x b, α x α b Then we hve the inequlity: 4) f x) dx α ) f ) + α α ) f x ) + b α ) f b) f q + ) p q α ) q+ + x α ) q+ + α x ) q+ + b α ) q+ q f q + ) p x ) q+ + b x ) q+ q q b )+ q q + ) q f p Proof Consider the division x 0 x x b nd the numbers α 0, α, x, α x, b, α 3 b Applying Theorem 43 for these prticulr choices, we esily obtin the desired inequlities We omit the detils Corollry 50 Let f be s bove nd x, b Then we hve Ostrowski s inequlity see lso 8): 5) f x) dx b ) f x ) f q + ) p x ) q+ + b x ) q+ q q b )+ q q + ) q f p The proof follows by the bove theorem choosing α, α b

43 35 SS Drgomir nd TM Rssis Remrk 4 If we choose x +b in 5), then we get the midpoint inequlity 8 ) + b 6) f x) dx f b ) b )+ q f q + ) p q The following corollry generlizing Simpson s inequlity holds s well: Corollry 5 Let f be s bove nd x 5+b 6, +5b 6 Then we hve the inequlity: f x) dx b f ) + f b) + f x ) 7) 3 f q + ) p q b )q+ 6 q+ + x 5 + b ) q+ ) q+ /q + 5b + x 6 6 The proof follows by Proposition 49 by choosing α 5+b nd α +5b Remrk 5 Now, if in 7) we choose x +b, then we get Simpson s inequlity 8 f x) dx b ) f ) + f b) + b 8) + f 3 q+ ) q + b ) + q f 6 q + ) q 3 p The following corollry lso holds 7 see lso 3): Corollry 5 Let f be s bove nd α +b α b Then we hve the inequlity: 9) f x) dx α ) f ) + α α ) f f q + ) p q α ) q+ + b + q + ) q f p b ) + q + b ) + b α ) f b) ) q+ α + α + b ) q+ q + b α ) q+ Finlly, we hve 7 see lso 3):

44 GENERALISATIONS OF OSTROWSKI INEQUALITY AND APPLICATIONS 36 Remrk 6 Now, if we choose in 9), α 3+b 4 nd α +3b 4, then we get the inequlity f x) dx ) f ) + f b) + b 0) + f 4 q + ) q f p b ) + q 44 Prticulr Qudrture Formule Let us consider the prtitioning of the intervl, b given by n : x 0 < x < < x n < x n b nd put h i : x i+ x i i 0,, n ) nd ν h) : mx {h i i 0,, n } The following theorem holds 7: Theorem 53 Let f :, b R be bsolutely continuous on, b nd k Then we hve the composite qudrture formul ) where ) A k n, f) : T n, f) + k nd f x) dx A k n, f) + R k n, f), n k k j) xi + jx i+ f k j 3) T n, f) : n f x i ) + f x i+ ) h i is the trpezoid qudrture formul The reminder R k n, f) stisfies the estimte h i 4) R k n, f) k q + ) q n f p h q+ i ) q, p >, p + q Proof Applying Corollry 45 on the intervls x i, x i+ i 0,, n ), we get xi+ f x) dx f x i ) + f x i+ ) h i + h k i k j) xi + jx i+ f x i k k k j xi+ ) f p t) dt k q + ) q h + q i x i

45 37 SS Drgomir nd TM Rssis Summing over i from 0 to n nd using the generlized tringle inequlity, we hve: R k n, f) n xi+ f x) dx x i k f x i) + f x i+ ) n h + q i k q + ) q n x i h i + h i k xi+ x i k k j) xi + jx i+ f k j f t) p dt By Hölder s discrete inequlity, we hve n h + xi+ ) q i f t) p p dt n ) q h + q q i h q+ i nd the theorem is proved The following corollry holds: ) q f p, n xi+ x i ) p ) ) p q f t) p p dt Corollry 54 Let f, n be s bove Then we hve the qudrture formul: 5) f x) dx T n, f) + M n, f) + R n, f), where M n, f) is the midpoint rule, nmely, n ) xi + x i+ M n, f) : f h i The reminder R n, f) stisfies the estimte: n 6) R n, f) f 4 q + ) p q The following corollry holds s well h q+ i ) q Corollry 55 Let f, n be s bove Then we hve the formul 7) f x) dx n ) n xi + x i+ T n, f) + f h i + f 3 3 +R 3 n, f) ) xi + x i+ h i 3

46 GENERALISATIONS OF OSTROWSKI INEQUALITY AND APPLICATIONS 38 The reminder R 3 n, f) stisfies the bound: 8) R 3 n, f) 6 q + ) q n f p h q+ i ) q The following theorem lso holds 7 see lso 4): Theorem 56 Let f nd n be s bove Suppose tht ξ i x i, x i+ i 0,, n ) Then we hve the formul: 9) n f x) dx ξ i x i ) f x i ) + x i+ ξ i ) f x i+ ) + R ξ, n, f) The reminder R ξ, n, f) stisfies the inequlity: 30) R ξ, n, f) q + ) q q + ) q n f p n f p n ξ i x i ) q+ + h q+ i ) q x i+ ξ i ) q+ q get Proof Apply Proposition 48 on the intervls x i, x i+ i 0,, n ) to xi+ f x) dx ξ i x i ) f x i ) + x i+ ξ i ) f x i+ ) x i ξ q + ) i x i ) q+ + x i+ ξ i ) q+ xi+ ) q f t) p p dt q x i

47 39 SS Drgomir nd TM Rssis Summing over i from 0 to n, using the generlized tringle inequlity nd Hölder s discrete inequlity, we my stte R ξ, n, f) n xi+ f x) dx ξ i x i ) f x i ) + x i+ ξ i ) f x i+ ) x i n { ξ i x i ) q+ + x i+ ξ i ) q+ q q + ) q xi+ x i f t) p dt n q + ) q n xi+ q + ) q x i n ) } p ξ i x i ) q+ + x i+ ξ i ) q+ ) q ) q q ) ) p p f t) p p dt nd the first inequlity in 30) is proved n q ξ i x i ) q+ + x i+ ξ i ) q+ f p The second inequlity is obvious by tking into ccount tht for ll i 0,, n ξ i x i ) q+ + x i+ ξ i ) q+ h q+ i The following corollry contins some prticulr well known qudrture formule: Corollry 57 Let f nd n be s bove Then we hve 3) 3) 33) ) The left rectngle rule ) The right rectngle rule 3) The trpezoid rule where n f x) dx f x i ) h i + R l n, f) ; n f x) dx f x i+ ) h i + R r n, f) ; f x) dx n f x i ) + f x i+ ) h i + R T n, f), n 34) R l n, f), R r n, f) f q + ) p q h q+ i ) q

48 GENERALISATIONS OF OSTROWSKI INEQUALITY AND APPLICATIONS 40 nd R T n, f) q + ) q n f p h q+ i ) q The following theorem holds s well 7 Theorem 58 Let f nd n be s bove If x i α ) i ξ i α ) i x i+ i 0,, n ), then we hve the formul: 35) n f x) dx n + α ) i x i ) f x i ) + x i+ α ) i n ) f x i+ ) + R ) α ) i α ) i f ξ i ) ) ξ, α ), α ), n, f, where the reminder stisfies the estimte R ξ, α ), α ) 36), n, f) n ) q+ n f q + ) p α ) i x i + ξ i α ) i q n ) q+ n + α ) i ξ i + q + ) q q + ) q n f p n f p x i+ α ) i n ξ i x i ) q+ + h q+ i ) q ) q+ q ) q+ x i+ ξ i ) q+ q Proof The proof follows by Proposition 49 pplied on the intervls x i, x i+ i 0,, n ) We omit the detils The following corollry of the bove theorem holds see lso 8) Corollry 59 Let f, n be s bove nd ξ i x i, x i+ i 0,, n ) Then we hve the formul of Riemnn s type: 37) n f x) dx f ξ i ) h i + R R ξ, n, f)

49 4 SS Drgomir nd TM Rssis The reminder R R ξ, n, f) stisfies the estimte 38) R R ξ, n, f) q + ) q q + ) q n f p n f p n ξ i x i ) q+ + h q+ i ) q x i+ ξ i ) q+ q Remrk 7 If we choose in 37), ξ i xi+xi+, then we get the midpoint qudrture formul where The following corollry lso holds f x) dx M n, f) + R M n, f), n R M n, f) f q + ) p q Corollry 60 Let f, n be s bove nd ξ i Then we hve the formul: 39) h q+ i ) q 5xi+x i+ 6, xi+5xi+ 6 i 0,, n ) f x) dx n f x i ) + f x i+ ) h i + n f ξ 6 3 i ) h i + R S ξ, n, f), where the reminder, R S ξ, n, f), stisfies the estimte: 40) R S ξ, n, f) n f q + ) p q 3 6 q h q+ i + n xi + 5x i+ + 6 ξ i ) q+ q n ξ i 5x ) q+ i + x i+ 6 Remrk 8 Now, if in 39) we choose ξ i xi+xi+, then we obtin Simpson s qudrture formul 4) n f x) dx f x i ) + f x i+ ) h i + n xi + x i+ f 6 3 where the reminder term R S n, f) stisfies the inequlity 8: q+ + 4) R S n, f) 6 q + ) q 3 The following corollry lso holds ) q ) h i + R S n, f), n f p h q+ i ) q

50 GENERALISATIONS OF OSTROWSKI INEQUALITY AND APPLICATIONS 4 Corollry 6 Let f, n be s bove nd x i α ) i i 0,, n ) Then we hve the formul xi+xi+ α ) i x i+ 43) n f x) dx n + α ) i x i ) f x i ) + x i+ α ) i n ) α ) i α ) i f ) xi + x i+ ) f x i+ ) + R B α ), α ), n, f ) The reminder stisfies the estimte R B α ), α ) 44), n, f) n ) q+ n f q + ) p α ) xi + x i+ i x i + q n + α ) i x ) q+ n i + x i+ + q + ) q n f p h q+ i ) q x i+ α ) i ) q+ q ) q+ α ) i Finlly, we hve Remrk 9 If we choose in 43), α ) i we get the formul: 3xi+xi+ 4 nd α ) i xi+3xi+ 4, then 45) f x) dx T n, f) + M n, f) + R B n, f) The reminder R B n, f) stisfies the bound: 46) R B n, f) 4 q + ) q n f p h q+ i ) q 5 Generlistions in Terms of L norm 5 Some Inequlities We strt with the following theorem 6 Theorem 6 Let I k : x 0 < x < < x k < x k b be division of the intervl, b nd α i i 0,, k + ) be k + points such tht α 0, α i x i, x i i,, k) nd α k+ b If f :, b R is bsolutely continuous

51 43 SS Drgomir nd TM Rssis on, b, then we hve the inequlity: k 47) f x) dx α i+ α i ) f x i ) { ν h) + mx α i+ x } i + x i+, i 0,, k f ν h) f, where ν h) : mx {h i i 0,, k }, h i : x i+ x i i 0,, k ) nd f : f t) dt, is the usul L, b norm 48) Proof Integrting by prts, we hve see lso Theorem ): f t) dt k α i+ α i ) f x i ) On the other hnd, we hve k K t) f t) dt However, xi+ x i t x i,x i+ t α i+ f t) dt sup Then T Now, s then xi+ K t) f t) dt x i t α i+ f t) dt : T xi+ t α i+ f t) dt x i xi+ mx {α i+ x i, x i+ α i+ } f t) dt x i x i+ x i ) + α i+ x i + x i+ xi+ f t) dt k h i + α i+ x i + x i+ xi+ f t) dt x i mx,,k h i + α i+ x i + x i+ k xi+ f t) dt x i { ν h) + mx α i+ x } i + x i+, i 0,, k f : V mx nd, consequently, α i+ x i + x i+ h i, { α i+ x i + x i+ The theorem is completely proved }, i 0,, k ν h) V ν h) f x i

52 GENERALISATIONS OF OSTROWSKI INEQUALITY AND APPLICATIONS 44 Now, if we ssume tht the points of the division I k re given, then the best inequlity we cn obtin from Theorem 43 is embodied in the following corollry Corollry 63 Let f nd I k be s bove Then we hve the inequlity: 49) f x) dx k x ) f ) + x i+ x i ) f x i ) + b x k ) f b) ν h) f i Proof The proof is obvious by the bove theorem nd we omit the detils The following corollry for equidistnt prtitioning is useful in prctice Corollry 64 Let I k : x i : + b ) i, i 0,, k) k be n equidistnt prtitioning of, b If f is s bove, then we hve the inequlity: b k f ) + f b) b ) k i) + ib 50) f x) dx b ) + f k k k k b ) f i 5 A Generl Qudrture Formul Let n : x n) 0 < x n) < < x n) n < xn) n b be sequence of divisions of, b nd consider the sequence of numericl integrtion formule see Subsection 3 nd 4) I n f, n, w n ) : n j0 w n) j f x n) j where w n) j j 0,, n) re the qudrture weights nd n j0 wn) j b The following theorem provides sufficient condition for the weights w n) j such tht I n f, n, w n ) pproximtes the integrl f x) dx with n error expressed in terms of f, 6 Theorem 65 Let f :, b R be n bsolutely continuous mpping on, b If the qudrture weights w n) j stisfy the condition x n) i i j0 w n) j ), x n) i+ for ll i 0,, n,

53 45 SS Drgomir nd TM Rssis then we hve the estimte I 5) n f, n, w n ) f x) dx h ν n)) i + mx + j0 ν h n)) f, w n) j xn) i + x n) i+, i 0,, n f where ν h n)) { } : mx h n) i i 0,, n nd h n) i : x n) i+ xn) i In prticulr, 5) lim I n f, n, w n ) νh n) ) 0 uniformly by the influence of the w n f x) dx Proof The proof is similr to tht of Theorem 46 nd we omit it Now, consider the equidistnt prtitioning of, b given by E n : x n) i : + i b ) i 0,, n) ; n nd define the sequence of numericl qudrture formule by I n f, w n ) : n w n) i f + in b ) The following corollry which cn be more useful in prctice holds: Corollry 66 Let f be s bove If the qudrture weights w n) j condition: 53) i n b i j0 then we hve: I 54) n f, w n ) f x) dx b n + mx i + j0 b ) f n In prticulr, we hve the limit w n) j i +, i 0,, n ; n w n) j i + lim I n f, w n ) n uniformly by the influence of the weights w n b ) n f x) dx, stisfy the, i 0,, n f

54 GENERALISATIONS OF OSTROWSKI INEQUALITY AND APPLICATIONS Prticulr Inequlities The following proposition holds 6 see lso 4) Proposition 67 Let f :, b R be n bsolutely continuous mpping on, b Then we hve the inequlity: 55) f x) dx α ) f ) + b α) f b) b ) + α + b f for ll α, b The proof follows by Theorem 43 choosing x 0, x b, α 0, α α, b nd α b Remrk 0 inequlity 56) ) If in 55) we put α b, then we get the left rectngle f x) dx b ) f ) b ) f ; b) If α, then, by 55), we get the right rectngle inequlity 57) f x) dx b ) f b) b ) f ; c) It is esy to see tht the best inequlity we cn get from 55) is for α +b obtining the trpezoid inequlity 4: f ) + f b) 58) f x) dx b ) b ) f Another proposition with mny interesting prticulr cses is the following one 6 see lso 3): Proposition 68 Let f be s bove nd x b, α x α b Then we hve 59) f x) dx α ) f ) + α α ) f x ) + b α ) f b) b ) + x + b + α + x + α x + b + α + x α x + b f b ) + x + b f b ) f Remrk If we choose bove α, α b, then we get the following Ostrowski s type inequlity obtined by Drgomir-Wng in the recent pper 7: 60) f x) dx b ) f x ) b ) + x + b f

55 47 SS Drgomir nd TM Rssis for ll x, b We note tht the best inequlity we cn get in 5) is for x +b obtining the midpoint inequlity see lso ) ) + b 6) f x) dx f b ) b ) f b) If we choose in 59) α 5+b 6, α +5b 6 nd x 5+b get f x) dx b f ) + f b) + f x ) 6) 3 b ) + x + b { + mx x + b } 3, + b x 3 6, +5b 6, then we In prticulr, if we choose in 6), x +b, then we get the following Simpson s inequlity 9 f x) dx b ) f ) + f b) + b 63) + f 3 3 b ) f 54 Prticulr Qudrture Formule Let us consider the prtitioning of the intervl, b given by n : x 0 < x < < x n < x n b nd put h i : x i+ x i i 0,, n ) nd ν h) : mx {h i i 0,, n } The following theorem holds 6: Theorem 69 Let f :, b R be bsolutely continuous on, b nd k Then we hve the composite qudrture formul 64) where 65) A k n, f) : T n, f) + k nd f x) dx A k n, f) + R k n, f), n k k j) xi + jx i+ f k j 66) T n, f) : n f x i ) + f x i+ ) h i is the trpezoid qudrture rule h i

56 GENERALISATIONS OF OSTROWSKI INEQUALITY AND APPLICATIONS 48 The reminder R k n, f) stisfies the estimte 67) R k n, f) k ν h) f Proof Applying Corollry 45 on the intervls x i, x i+ i 0,, n ), we obtin xi+ f x) dx x i k f x i) + f x i+ ) h i + h k i k j) xi + jx i+ f k k k h i xi+ x i f t) dt Now, using the generlized tringle inequlity, we get: R k n, f) n xi+ f x) dx x i k f x i) + f x i+ ) h i + h i k n xi+ h i k x i nd the theorem is proved f t) dt ν h) k j k k j) xi + jx i+ f k j n xi+ x i f t) dt ν h) k f, The following corollries hold: Corollry 70 Let f be s bove Then we hve the formul: 68) f x) dx T n n, f) + M n n, f) + R n, f), where M n n, f) is the midpoint qudrture formul, n ) xi + x i+ M n n, f) : f h i nd the reminder R n, f) stisfies the inequlity: 69) R n, f) 4 ν h) f Corollry 7 Under the bove ssumptions, we hve 70) f x) dx n ) n xi + x i+ T n n, f) + f h i + f 3 3 +R 3 n, f), ) xi + x i+ h i 3

57 49 SS Drgomir nd TM Rssis where the reminder, R 3 n, f), stisfies the bound: 7) R 3 n, f) 6 ν h) f The following theorem holds 6 see lso 4): Theorem 7 Let f nd n be s bove nd ξ i x i, x i+ i 0,, n ) Then we hve the qudrture formul: 7) n f x) dx ξ i x i ) f x i ) + x i+ ξ i ) f x i+ ) + R ξ, n, f), where the reminder, R ξ, n, f), stisfies the estimtion: 73) R ξ, n, f) { ν h) + mx ξ i x } i + x i+, i 0,, n f ν h) f for ll ξ i s bove Proof Follows by Proposition 48 nd we omit the detils Corollry 73 Let f nd n be s bove Then we hve ) The left rectngle rule 74) 75) n f x) dx f x i ) h i + R l n, f) ; ) The right rectngle rule n f x) dx f x i+ ) h i + R r n, f) ; 3) The trpezoid rule 76) f x) dx T n, f) + R T n, f), where nd R l n, f) R r n, f) ν h) f R T n, f) ν h) f The following theorem lso holds 6

58 GENERALISATIONS OF OSTROWSKI INEQUALITY AND APPLICATIONS 50 Theorem 74 Let f nd n be s bove nd ξ i x i, x i+, x i α ) i x i+, then we hve the qudrture formul: α ) i 77) n f x) dx n + α ) i x i ) f x i ) + x i+ α ) i n ) f x i+ ) + R ) α ) i α ) i f ξ i ) ) ξ, α ), α ), n, f, ξ i where the reminder, R ξ, α ), α ), n, f ), stisfies the estimte R ξ, α ), α ) 78), n, f) { ν h) + mx,,n ξ i x i + x i+ { + mx mx,,n α) i x i + ξ i, mx,,n α) i ξ }} i + x i+ f ν h) + mx,n ξ i x i + x i+ f ν h) f Proof Follows by Proposition 49 nd we omit the detils The following corollry is the result of Drgomir-Wng from the recent pper 7 Corollry 75 Under the bove ssumptions, we hve the Riemnn s qudrture formul: 79) n f x) dx f ξ i ) h i + R R ξ, n, f) The reminder R R ξ, n, f) stisfies the bound 80) R R ξ, n, f) { ν h) + mx ξ i x } i + x i+, i 0,, n f ν h) f for ll ξ i x i, x i+ i 0,, n) Finlly, the following corollry which generlizes Simpson s qudrture formul holds Corollry 76 Under the bove ssumptions nd if ξ i xi++5x i 6, xi+5xi+ 6 i 0,, n ), then we hve the formul: 8) f x) dx n f x i ) + f x i+ ) h i + n f ξ 6 3 i ) h i + S f, n, ξ),

59 5 SS Drgomir nd TM Rssis where the reminder, S f, n, ξ), stisfies the estimte: 8) S f, n, ξ) { { ν h) + mx ξ i x } i + x i+,,n { + mx mx,,n ξ i x i + x i+ 3, mx,,n x i + x i+ 3 The proof follows by the inequlity 6) nd we omit the detils ξ i }} f Remrk Now, if we choose in 8), ξ i xi+xi+, then we get Simpson s qudrture formul 9 83) f x) dx n f x i ) + f x i+ ) h i 6 + n ) xi + x i+ f h i + S f, n ), 3 where the reminder term S f, n ) stisfies the bound: 84) S f, n ) 3 ν h) f

60

61 Bibliogrphy GA ANASTASSIOU, Ostrowski type inequlities, Proc of Amer Mth Soc, 3) 999), P CERONE nd SS DRAGOMIR, Midpoint-type rules from n inequlities point of view, Hndbook of Anlytic-Computtionl Methods in Applied Mthemtics, Editor: G Anstssiou, CRC Press, New York 000), P CERONE nd SS DRAGOMIR, Three point qudrture rules involving, t most, first derivtive, submitted) ON LINE: 4 P CERONE nd SS DRAGOMIR, Trpezoidl-type rules from n inequlities point of view, Hndbook of Anlytic-Computtionl Methods in Applied Mthemtics, Editor: G Anstssiou, CRC Press, New York 000), SS DRAGOMIR, A generliztion of Ostrowski integrl inequlity for mppings whose derivtives belong to L, b nd pplictions in numericl integrtion, SUT J Mth, ccepted) 6 SS DRAGOMIR, A generliztion of Ostrowski integrl inequlity for mppings whose derivtives belong to L, b nd pplictions in numericl integrtion, J of Computtionl Anlysis nd Applictions, in press) 7 SS DRAGOMIR, A generliztion of Ostrowski integrl inequlity for mppings whose derivtives belong to L p, b, < p <, nd pplictions in numericl integrtion, J Mth Anl Appl, in press) 8 SS DRAGOMIR, On Simpson s qudrture formul for differentible mppings whose derivtives belong to L p spces nd pplictions, J KSIAM, 998), SS DRAGOMIR, On Simpson s qudrture formul for mppings of bounded vrition nd pplictions, Tmkng J of Mth, 30 )999), SS DRAGOMIR, On Simpson s qudrture formul for Lipschitzin mppings nd pplictions, Soochow J of Mth, 5 )999), SS DRAGOMIR, On the Ostrowski s integrl inequlity for mppings with bounded vrition nd pplictions, Mth Ineq & Appl, ccepted) ON LINE: SS DRAGOMIR, Ostrowski inequlity for monotonous mppings nd pplictions, J KSIAM, 3) 999), SS DRAGOMIR, The Ostrowski s integrl inequlity for Lipschitzin mppings nd pplictions, Comp nd Mth with Appl, ), SS DRAGOMIR, The Ostrowski s integrl inequlity for mppings of bounded vrition, Bull Austrl Mth Soc, ), SS DRAGOMIR, P CERONE, J ROUMELIOTIS nd S WANG, A weighted version of Ostrowski inequlity for mppings of Hölder type nd pplictions in numericl nlysis, Bull Mth Soc Sc Mth Roumnie, 490)4) 99), SS DRAGOMIR, J PEČARIĆ nd S WANG, The unified tretment of trpezoid, Simpson nd Ostrowski type inequlities for monotonic mppings nd pplictions, Mth nd Comp Modelling, 3 000), SS DRAGOMIR nd S WANG, A new inequlity of Ostrowski s type in L norm nd pplictions to some specil mens nd to some numericl qudrture rules, Tmkng J of Mth, 8 997), SS DRAGOMIR nd S WANG, A new inequlity of Ostrowski s type in L p norm nd pplictions to some specil mens nd to some numericl qudrture rules, Indin Journl of Mthemtics, 403) 998),

62 BIBLIOGRAPHY 54 9 SS DRAGOMIR nd S WANG, An inequlity of Ostrowski-Grüss type nd its pplictions to the estimtion of error bounds for some specil mens nd for some numericl qudrture rules, Computers Mth Applic, ), SS DRAGOMIR nd S WANG, Applictions of Ostrowski s inequlity to the estimtion of error bounds for some specil mens nd some numericl qudrture rules, Appl Mth Lett, 998), AM FINK, Bounds on the derivtion of function from its verges, Czech Mth J, 4 99), DS MITRINOVIĆ, JE PE CARIĆ nd AM FINK, Inequlities Involving Functions nd Their Integrls nd Derivtives, Kluwer Acdemic Publishers, Dordrecht, 994

63 CHAPTER Integrl Inequlities for n Times Differentible Mppings by A SOFO Abstrct This chpter investigtes generlistions of Integrl Inequlities for n-times differentible mppings With the id of the modern theory of inequlities nd by the use of generl Peno kernel, explicit bounds for interior point rules re obtined Integrl equlities re obtined which re then used to obtin inequlities for n- times differentible mppings on the three norms, p nd Some prticulr inequlities re investigted which include explicit bounds for perturbed trpezoid, midpoint, Simpson s, Newton-Cotes nd left nd right rectngle rules The inequlities re lso pplied to vrious composite qudrture rules nd the nlysis llows the determintion of the prtition required tht would ssure tht the ccurcy of the result would be within prescribed error tolernce Introduction In 938 Ostrowski 9 obtined bound for the bsolute vlue of the difference of function to its verge over finite intervl The theorem is s follows Theorem Let f :, b R be differentible mpping on, b nd let f t) M for ll t, b), then the following bound is vlid ) f x) b ) f t) dt b x +b 4 + b ) b ) M for ll x, b The constnt 4 is shrp in the sense tht it cnnot be replced by smller one Drgomir nd Wng 9, 0,, extended the result ) nd pplied the extended result to numericl qudrture rules nd to the estimtion of error bounds for some specil mens 55

64 INTEGRAL INEQUALITIES FOR n TIMES DIFFERENTIABLE MAPPINGS 56 Drgomir 4, 5, 6 further extended the result ) to incorporte mppings of bounded vrition, Lipschitzin mppings nd monotonic mppings Cerone, Drgomir nd Roumeliotis 6 s well s Dedić, Mtić nd Pečrić 8 nd Perce, Pečrić, Ujević nd Vrošnec 30 further extended the result ) by considering n times differentible mppings on n interior point x, b Cerone nd Drgomir, hve subsequently given number of other trpezoidl nd midpoint rules bsed on the Peno kernel pproch Drgomir 9, 0,, further refined the inequlity ) by considering n intervl, b with multiple number of subdivisions In this current work we extend, subsume nd generlise some previous results, by considering n times differentible mppings We investigte interior point rules by tking into considertion multiple subdivisions of n intervl, b Moreover, we obtin explicit bounds through the use of Peno kernel pproch nd the modern theory of inequlities This pproch lso permits the investigtion of qudrture rules tht plce fewer restrictions on the behviour of the integrnd nd thus dmits lrger clss of functions In Section we develop number of integrl identities, for n time differentible mppings, which re of interest in themselves nd utilise them, in Section 3, to obtin integrl inequlities on the Lebesgue spces, L, b, L p, b nd L, b In Section 4 we investigte the convergence of generl qudrture formul tht permits the pproximtion of the integrl of function over finite intervl In Section 5 we employ the pre-grüss reltionship to obtin more integrl inequlities for n times differentible mppings In Section 6 we point out number of prticulr specil cses tht incorporte the generlised left nd right rectngle inequlities, the perturbed trpezoid nd midpoint inequlities, Simpson s inequlity, the generlised Newton-Cotes three eighths inequlity nd Boole type reltionship Finlly, in Section 7, we pply some of the inequlities to numericl qudrture rules Integrl Identities Theorem Let I k : x 0 < x < < x k < x k b be division of the intervl, b nd α i i 0,, k + ) be k + points so tht α 0, α i x i, x i i,, k) nd α k+ b If f :, b R is mpping such tht f n ) is bsolutely continuous on, b, then for ll x i, b we hve the identity: n ) j k { ) f t) dt + x i+ α i+ ) j f j ) x i+ ) j! j } x i α i+ ) j f j ) x i ) ) n K n,k t) f n) t) dt,

65 57 A sofo where the Peno kernel t α ) n, t, x ) n! t α ) n, t x, x ) n! 3) K n,k t) : t α k ) n, t x k, x k ) n! t α k ) n, t x k, b, n! n nd k re nturl numbers, n, k nd f 0) x) f x) Proof The proof is by mthemticl induction For n, from ) we hve the equlity 4) where f t) dt k x i+ α i+ ) j f x i+ ) x i α i+ ) j f x i ) K,k t) f t) dt, t α ), t, x ) K,k t) : t α ), t x, x ) t α k ), t x k, x k ) t α k ), t x k, b To prove 4), we integrte by prts s follows k k K,k t) f t) dt t α i+ ) f t) xi+ x i xi+ x i xi+ x i t α i+ ) f t) dt f t) dt k k x i+ α i+ ) f x i+ ) x i α i+ ) f x i ) k f t) dt + K,k t) f t) dt xi+ x i+ α i+ ) f x i+ ) x i α i+ ) f x i ) Hence 4) is proved x i f t) dt

66 INTEGRAL INEQUALITIES FOR n TIMES DIFFERENTIABLE MAPPINGS 58 Assume tht ) holds for n nd let us prove it for n + We need to prove the equlity 5) k n+ ) j f t) dt + j! j } x i α i+ ) j f j ) x i ) ) n+ K n+,k t) f n+) t) dt, { x i+ α i+ ) j f j ) x i+ ) where from 3) K n+,k t) : t α ) n+, t, x ) n + )! t α ) n+, t x, x ) n + )! t α k ) n+, t x k, x k ) n + )! t α k ) n+, t x k, b n + )! Consider k K n+,k t) f n+) t) dt xi+ x i n+ t α i+) f n+) t) dt n + )! nd upon integrting by prts we hve K n+,k t) f n+) t) dt k t α i+ ) n+ f n) t) n + )! { k x i+ α i+ ) n+ n + )! K n,k t) f n) t) dt x i+ x i xi+ x i n t α i+) f n) t) dt n! } f n) x i+ ) x i α i+ ) n+ f n) x i ) n + )!

67 59 A sofo Upon rerrngement we my write K n,k t) f n) t) dt { x i+ α i+ ) n+ k n + )! K n+,k t) f n+) t) dt } f n) x i+ ) x i α i+ ) n+ f n) x i ) n + )! Now substitute K n,k t) f n) t) dt from the induction hypothesis ) such tht k n ) n f t) dt + ) n ) j { x i+ α i+ ) j j! j } f j ) x i+ ) x i α i+ ) j f j ) x i ) { } k x i+ α i+ ) n+ f n) x i+ ) x i α i+ ) n+ f n) x i ) n + )! n + )! K n+,k t) f n+) t) dt Collecting the second nd third terms nd rerrnging, we cn stte k n+ ) j f t) dt + j! j } x i α i+ ) j f j ) x i ) ) n+ K n+,k t) f n+) t) dt, which is identicl to 5), hence Theorem is proved { x i+ α i+ ) j f j ) x i+ ) The following corollry gives slightly different representtion of Theorem, which will be useful in the following work Corollry 3 From Theorem, the equlity ) my be represented s 6) f t) dt + n ) j k { x i α i ) j x i α i+ ) j} f j ) x i ) j! j ) n K n,k t) f n) t) dt

68 INTEGRAL INEQUALITIES FOR n TIMES DIFFERENTIABLE MAPPINGS 60 Proof From ) consider the second term nd rewrite it s 7) k S + S : { x i+ α i+ ) f x i+ ) + x i α i+ ) f x i )} k n + ) j j! j x i α i+ ) j f j ) x i )} { x i+ α i+ ) j f j ) x i+ ) Now k S α ) f ) + x i α i+ ) f x i ) k + i { x i+ α i+ ) f x i+ )} b α k ) f b) k α ) f ) + x i α i+ ) f x i ) k + i { x i α i ) f x i )} b α k ) f b) i k α ) f ) α i+ α i ) f x i ) b α k ) f b) i Also, S k n ) j j! + j n ) j j j! n ) j j! j k n ) j j! i j { x i+ α i+ ) j f j ) x i+ )} { } x k α k ) j f j ) x k ) { } x 0 α ) j f j ) x 0 ) { x i α i+ ) j f j ) x i )} n ) j n b α k ) j f j ) ) j b) α ) j f j ) ) j! j! j j k n + ) j { x i α i ) j x i α i+ ) j} f j ) x i ) j! i j

69 6 A sofo From 7) { } k S + S : b α k ) f b) + α ) f ) + α i+ α i ) f x i ) n ) j { } + b α k ) j f j ) b) α ) j f j ) ) j! j k n + ) j { x i α i ) j x i α i+ ) j} f j ) x i ) j! i j { } k α ) f ) + α i+ α i ) f x i ) + b α k ) f b) i i n ) j + α ) j f j ) ) j! j k { + x i α i ) j x i α i+ ) j} f j ) x i ) + b α k ) j f j ) b) Keeping in mind tht x 0, α 0 0, x k b nd α k+ b we my write S + S k α i+ α i ) f x i ) n ) j k + j! j n ) j k j j! i { x i α i ) j x i α i+ ) j} f j ) x i ) { x i α i ) j x i α i+ ) j} f j ) x i ) And substituting S +S into the second term of ) we obtin the identity 6) If we now ssume tht the points of the division I k re fixed, we obtin the following corollry Corollry 4 Let I k : x 0 < x < < x k < x k b be division of the intervl, b If f :, b R is s defined in Theorem, then we hve the equlity 8) f t) dt + n j j j! k ) n K n,k t) f n) t) dt, where h i : x i+ x i, h : 0 nd h k : 0 { } h j i + )j h j i f j ) x i )

70 INTEGRAL INEQUALITIES FOR n TIMES DIFFERENTIABLE MAPPINGS 6 Proof Choose α 0, α + x, α x + x,, α k x k + x k, α k x k + x k From Corollry 3, the term the term nd the term nd α k+ b k b α k ) f b) + α ) f ) + α i+ α i ) f x i ) i { } k h 0 f ) + h i + h i ) f x i ) + h k f b), n ) j j n j j! i { } b α k ) j f j ) b) α ) j f j ) ) ) j { } j! j h j k f j ) b) ) j h j 0 f j ) ) k n ) j { x i α i ) j x i α i+ ) j} f j ) x i ) j! i j k n ) j { } j! j h j i )j h j i f j ) x i ) i j Putting the lst three terms in 6) we obtin { } b f t) dt k h 0 f ) + h i + h i ) f x i ) + h k f b) i n ) j { } + j! j h j k f j ) b) ) j h j 0 f j ) ) j k n + ) j { } j! j h j i )j h j i f j ) x i ) i j ) n K n,k t) f n) t) dt Collecting the inner three terms of the lst expression, we hve n ) j k { } f t) dt + j! j h j i )j h j i f j ) x i ) j ) n K n,k t) f n) t) dt, which is equivlent to the identity 8)

71 63 A sofo The cse of equidistnt prtitioning is importnt in prctice, nd with this in mind we obtin the following corollry Corollry 5 Let ) b 9) I k : x i + i, i 0,, k k be n equidistnt prtitioning of, b, then we hve the equlity n ) j b 0) f t) dt + f j ) ) k j! j k { } + ) j f j ) x i ) + ) j f j ) b) i ) n K n,k t) f n) t) dt It is of some interest to note tht the second term of 0) involves only even derivtives t ll interior points x i, i,, k Proof Using 9) we note tht h 0 x x 0 b k, h k x k x k ) b k, h i x i+ x i b k nd h i x i x i b, i,, k ) k nd substituting into 8) we hve b n ) { j k b b f t) dt + j! j f j ) ) + k k j ) } j ) j + ) j b f j ) x i ) + ) j b f b) j ) k k ) n K n,k t) f n) t) dt, which simplifies to 0) fter some minor mnipultion ) j The following Tylor-like formul with integrl reminder lso holds Corollry 6 Let g :, y R be mpping such tht g n) is bsolutely continuous on, y Then for ll x i, y we hve the identity ) g y) k n g ) ) j j! j { x i+ α i+ ) j g j) x i+ ) } y x i α i+ ) j g j) x i ) + ) n K n,k y, t) g n+) t) dt

72 INTEGRAL INEQUALITIES FOR n TIMES DIFFERENTIABLE MAPPINGS 64 or ) g y) g ) n ) j k { x i α i ) j x i α i+ ) j} g j) x i ) j! j y + ) n K n,k y, t) g n+) t) dt The proof of ) nd ) follows directly from ) nd 6) respectively upon choosing b y nd f g 3 Integrl Inequlities In this section we utilise the equlities of Section nd develop inequlities for the representtion of the integrl of function with respect to its derivtives t multiple number of points within some intervl In prticulr, we develop inequlities which depend on the spces L, b, L p, b nd L, b Theorem 7 Let I k : x 0 < x < < x k < x k b be division of the intervl, b nd α i i 0,, k + ) be k + points so tht α 0, α i x i, x i i,, k) nd α k+ b If f :, b R is mpping such tht f n ) is bsolutely continuous on, b, then for ll x i, b we hve the inequlity: 3) f t) dt + f n) n + )! n ) j k { ) x i α i ) j x i α i+ ) j} f j ) x i j! j k {α i+ x i ) n+ + x i+ α i+ ) n+} f n) k n + )! h n+ i f n) n + )! b ) νn h) if f n) L, b, where f n) : sup f n) t) <, t,b h i : x i+ x i nd ν h) : mx {h i i 0,, k }

73 65 A sofo Proof From Corollry 3 we my write b n ) j k { ) f t) dt + x i α i ) j x i α i+ ) j} 4) f j ) x i j! j )n K n,k t) f n) t) dt, nd )n K n,k t) f n) f t) dt n) K n,k t) dt, nd thus K n,k t) dt k xi+ x i n t α i+ dt n! k αi+ α i+ t) x i n! n + )! n xi+ dt + α i+ n t α i+) dt n! k {α i+ x i ) n+ + x i+ α i+ ) n+} )n K n,k t) f n) t) dt f n) k { α i+ x i ) n+ + x i+ α i+ ) n+} n + )! Hence, from 4), the first prt of the inequlity 3) is proved The second nd third lines follow by noting tht f n) k { α i+ x i ) n+ + x i+ α i+ ) n+} f n) k h n+ i, n + )! n + )! since for 0 < B < A < C it is well known tht 5) A B) n+ + C A) n+ C B) n+ Also f n) k n + )! h n+ i f n) k n + )! νn h) h i f n) n + )! b ) νn h), where ν h) : mx {h i i 0,, k } nd therefore the third line of the inequlity 3) follows, hence Theorem 7 is proved Theorem 8 Let I k : x 0 < x < < x k < x k b be division of the intervl, b nd α i i 0,, k + ) be k + points so tht α 0, α i x i, x i i,, k) nd α k+ b If f :, b R is mpping such

74 INTEGRAL INEQUALITIES FOR n TIMES DIFFERENTIABLE MAPPINGS 66 tht f n ) is bsolutely continuous on, b, then for ll x i, b we hve the inequlity: b n 6) ) j k { ) f t) dt + x i α i ) j x i α i+ ) j} f j ) x i j! f n) p n! nq + ) q j f n) p n! nq + ) q k k { α i+ x i ) nq+ + x i+ α i+ ) nq+}) q h nq+ i ) q f n) ) p b ν n q h) n! nq + if f n) L p, b nd p >, p + q, where f n) p : f n) t) p dt ) p Proof From Corollry 3 we my use 4) nd by Hölder s integrl inequlity write b ) b q 7) )n K n,k t) f n) f t) dt n) K n,k t) q dt, p K n,k t) q dt From 7) k αi+ nq α i+ t) x i n!) q dt + n!) q nq + ) K n,k t) q dt n! nq + ) q k ) q xi+ α i+ nq t α i+) n!) q dt k {α i+ x i ) nq+ + x i+ α i+ ) nq+} { α i+ x i ) nq+ + x i+ α i+ ) nq+}) q nd utilising 4) we obtin the first line of the inequlity 6) Using the inequlity 4) we know tht k { α i+ x i ) nq+ + x i+ α i+ ) nq+} k h nq+ i

75 67 A sofo nd the second line of the inequlity 6) follows From 6) f n) p n! nq + ) q k h nq+ i ) q f n) p ν n h) n! nq + ) q f n) p ν n h) n! k b nq + h i ) q ) q nd the third line in the inequlity 6) follows, hence Theorem 8 is proved Theorem 9 Let I k : x 0 < x < < x k < x k b be division of the intervl, b nd α i i 0,, k + ) be k + points so tht α 0, α i x i, x i i,, k) nd α k+ b If f :, b R is mpping such tht f n ) is bsolutely continuous on, b, then for ll x i, b we hve the inequlity: 8) b n ) j k { ) f t) dt + x i α i ) j x i α i+ ) j} f j ) x i j! j f n) ν h) n, if f n) L, b, n! where f n) f n) t) dt Proof From 4) 9) )n K n,k t) f n) t) dt f n) K n,k t) f n) sup K n,k t), t,b

76 INTEGRAL INEQUALITIES FOR n TIMES DIFFERENTIABLE MAPPINGS 68 where K n,k t) t α i+ ) n n! sup t x i,x i+ t α i+ ) n n! mx n! { α i+ x i ) n, x i+ α i+ ) n },,k n mx n! {α i+ x i ), x i+ α i+ )},,k { xi+ x i mx + n!,,k α i+ x } i+ + x i n { } n hi mx + mx n!,,k δ i,,k { } { } n hi hi mx + mx n!,,k,,k since δ i : α i+ x i+ + x i nd therefore δ i h i ) n mx n! {h i},,k n! ν h)n, hence the proof of inequlity 8) follows When the points of the division I k re fixed, we obtin the following inequlity Corollry 0 Let f nd I k be defined s in Corollry 4, then b n ) j k { } ) 0) f t) dt + j h j i j! + )j h j i+ f j ) x i j f n) k n + )! n h n+ i if f n) L, b, f n) k ) p q h nq+ n! nq + ) q n i if f n) L p, b, p > nd p + q, f n) ν h) n if f n) L, b n! Proof from Corollry 4 we choose α 0, α + x,, α k x k + x k, α k x k + x k nd α k+ b Now utilising the first line of the inequlity 3), we my evlute k { α i+ x i ) n+ + x i+ α i+ ) n+} k ) n+ hi

77 69 A sofo nd therefore the first prt of the inequlity 0) follows From 6) nd 8) we obtin the second nd third lines of 0), hence the corollry is proved For the equidistnt prtitioning cse we hve the following inequlity Corollry Let f be s defined in Theorem nd let I k be defined by 9) Then n ) j b ) f t) dt + k j! j k { } f j ) ) + ) j f j ) x i ) + ) j f b) j ) i f n) n + )! k) n b ) n+ if f n) L, b, f n) p n! nq + ) q k) n b )n+ q if f n) L p, b, f n) n! p > nd p + q ), n b if f n) L, b k Proof We my utilise 0) nd from 3), 6) nd 8) note tht h 0 x x 0 b k in which cse ) follows nd h i x i+ x i b k, i,, k The following inequlities for Tylor-like expnsions lso hold Corollry Let g be defined s in Corollry 6 Then we hve the inequlity ) g y) g ) + g n+) n + )! g n+) p k n ) j k { ) x i α i ) j x i α i+ ) j} g j) x i j! j { α i+ x i ) n+ + x i+ α i+ ) n+} if g n+) L, b k { α i+ x i ) nq+ + x i+ α i+ ) nq+}) q n! nq + ) q if g n+) L p, b, p > nd p + q, g n+) ν h) n if g n+) L, b n!

78 INTEGRAL INEQUALITIES FOR n TIMES DIFFERENTIABLE MAPPINGS 70 for ll x i, y where ν h) : mx {h i i 0,, k }, h i : x i+ x i, g n+) : sup g n+) t) <, g n+) p : g n+) : t,y y g n+) t) p dt y g n+) t) dt ) p nd Proof Follows directly from ) nd using the norms s in 3), 6) nd 8) When the points of the division I k re fixed we obtin the following Corollry 3 Let g be defined s in Corollry 6 nd I k : x 0 < x < < x k < x k y be division of the intervl, y Then we hve the inequlity 3) n g y) g ) + j g n+) k n + )! n j j! k { } ) h j i + )j h j i+ g j) x i h n+ i if g n+) L, y g n+) k ) p q h nq+ n! nq + ) q n i if g n+) L p, y, p > nd p + q, g n+) ν h) n if g n+) L, y n! Proof The proof follows directly from using 8) with the pproprite norms For the equidistnt prtitioning cse we hve: Corollry 4 Let g be defined s in Corollry 6 nd ) y I k : x i + i, i 0,, k k

79 7 A sofo be n equidistnt prtitioning of, y, then we hve the inequlity: n ) j y 4) g y) g ) + k j! j k { } g j) ) + ) j g j) x i ) + ) j g y) j) i g n+) n + )! k) n y ) n+ if g n+) L, y, g n+) p n! nq + ) q k) n y )n+ q if g n+) L p, y, g n+) n! p > nd p + q ), n y if g n+) L, y k Proof The proof follows directly upon using 0) with f g nd b y 4 The Convergence of Generl Qudrture Formul Let m : x m) 0 < x m) < < x m) m < xm) m b be sequence of division of, b nd consider the sequence of rel numericl integrtion formul ) 5) I m f, f,, f n), m, w m : m j0 w m) j f x m) j x m) j j s0 ) w m) s { n ) r m r! r ) r} j0 x m) j ) f r ) x m) j, j s0 w m) s where w j j 0,, m) re the qudrture weights nd ssume tht m j0 wm) j b The following theorem contins sufficient condition for the weights w m) j so tht ) I m f, f,, f n) b, m, w m pproximtes the integrl f x) dx with n error expressed in terms of f n) Theorem 5 Let f :, b R be continuous mpping on, b If the qudrture weights, w m) j stisfy the condition i 6) x m) i w m) j x m) i+ for ll i 0,, m j0 ) r

80 INTEGRAL INEQUALITIES FOR n TIMES DIFFERENTIABLE MAPPINGS 7 then we hve the estimtion ) 7) I m f, f,, f n), m, w m Also f n) n + )! f n) n + )! m m + h m) i i j0 ) n+ w m) j x m) i ) n+ f t) dt f n) ) ν h m) b ), where f n) L, b n + )! In prticulr, if f n) <, then ν h m)) { : mx,,m h m) i : x m) i+ xm) i x m) i+ i h m) i ) lim I m f, f,, f n), m, w m νh m) ) 0 uniformly by the influence of the weights w m } j0 f t) dt w m) j ) n+ Proof Define the sequence of rel numbers Note tht α m) i+ : + i α m) j0 i+ + m j0 By the ssumption 6), we hve α m) i+ x m) i, x m) i+ Define α m) 0 nd compute nd α m) α m) 0 w m) 0 α m) i+ αm) i + i j0 w m) j α m) m+ αm) m + w m) j, i 0,, m w m) j + b b for ll i 0,, m i m j0 w m) j j0 w m) j m w m) i i 0,, m ) j0 w m) j w m) m

81 73 A sofo Consequently nd let m j0 m w m) j f x m) j ) α m) i+ αm) i f x m) j ) j s0 x m) i ) m { n ) r m r! r w m) s ) r} : I m f, f,, f n), m, w m ) j0 w m) i f x m) j ) f r ) x m) j Applying the inequlity 3) we obtin the estimte 7) x m) i ), j s0 w m) s ) r On the L p norm the following theorem holds Theorem 6 Let the conditions of Theorem 5 pply Then we hve the estimtion ) 8) I m f, f,, f n), m, w m f t) dt f n) p n! nq + ) q f n) p n! nq + ) q f n) p n! m m + + h m) i i j0 w m) j x m) i x m) i+ i ) nq+ ) q )) ) ν h m) n b q, nq + j0 ) nq+ w m) j ) nq+ where f n) L p, b, p > nd p + q In prticulr, if f n) p <, then ) lim I m f, f,, f n), m, w m νh m) ) 0 uniformly by the influence of the weights w m f t) dt q The proof of this theorem follows the sme pttern s tht of Theorem 5 nd will not be given here Similrly on the L norm the following theorem lso holds

82 INTEGRAL INEQUALITIES FOR n TIMES DIFFERENTIABLE MAPPINGS 74 Theorem 7 Let the conditions of Theorem 5 pply Then we hve the estimtion ) 9) I m f, f,, f n) f n), m, w m f t) dt ν h m)) n, n! where f n) L, b In prticulr, if f n) <, then ) lim I m f, f,, f n), m, w m νh m) ) 0 uniformly by the influence of the weights w m f t) dt The proof of Theorem 7 follows the sme pttern s Theorem 5, nd will not be given here The cse when the prtitioning is equidistnt is importnt in prctice Consider the equidistnt prtition E m : x m) i : + i b, i 0,, m) m nd define the sequence of numericl qudrture formule : I m f, f,, f n), m, w m ) m j0 w m) j f + j b ) n j s0 ) j b ) n w m) s n ) r m r! r ) r} f r ) + j0 j b ) n { j b ) n ) j s0 w m) s ) r The following corollry holds Corollry 8 Let f :, b R be bsolutely continuous on, b qudrture weights w m) j stisfy the condition If the i m b i j0 then the following bound holds: I m w m) j i +, i 0,,, n, m f, f,, f n), m, w m ) f t) dt

83 75 A sofo f n) n + )! m i j0 ) ) n+ b i m ) b i + ) m w m) j i j0 w m) j ) n+ f n) n + )! f n) p n! nq + ) q f n) p n! nq + ) q ) n+ b, where f n) L, b, m m i j0 w m) j ) ) nq+ b i m ) b + i + ) m i j0 w m) j ) nq+ ) n+ b q, m nq + ) where f n) L p, b, p > nd p + q, f n) ) ν h m) n f n) n! n! ) n b, where f n) L, b, m q In prticulr, if f n),p, <, then ) lim I m f, f,, f n), w m m uniformly by the influence of the weights w m f t) dt The proof of Corollry 8 follows directly from Theorem 5 5 Grüss Type Inequlities The Grüss inequlity 5, is well known in the literture It is n integrl inequlity which estblishes connection between the integrl of product of two functions nd the product of the integrls of the two functions Theorem 9 Let h, g :, b R be two integrble functions such tht φ h x) Φ nd γ g x) Γ for ll x, b, φ, Φ, γ nd Γ re constnts Then we hve 30) T h, g) Φ φ) Γ γ), 4

84 INTEGRAL INEQUALITIES FOR n TIMES DIFFERENTIABLE MAPPINGS 76 where 3) T h, g) : b b h x) g x) dx h x) dx b g x) dx nd the inequlity 30) is shrp, in the sense tht the constnt 4 cnnot be replced by smller one For simple proof of this fct s well s generlistions, discrete vrints, extensions nd ssocited mteril, see 8 The Grüss inequlity is lso utilised in the ppers, 3, 8, 4 nd the references contined therein A premture Grüss inequlity is the following Theorem 0 Let f nd g be integrble functions defined on, b nd let γ g x) Γ for ll x, b Then 3) T h, g) Γ γ T f, f)), where T f, f) is s defined in 3) Theorem 0 ws proved in 999 by Mtić, Pečrić nd Ujević 7 nd it provides shrper bound thn the Grüss inequlity 30) The term premture is used to highlight the fct tht the result 3) is obtined by not fully completing the proof of the Grüss inequlity The premture Grüss inequlity is completed if one of the functions, f or g, is explicitly known We now give the following theorem bsed on the premture Grüss inequlity 3) Theorem Let I k : x 0 < x < < x k < x k b be division of the intervl, b, α i i 0,, k + ) be k + points such tht α 0, α i x i, x i i,, k) nd α k b If f :, b R is bsolutely continuous nd n time differentible on, b, then ssuming tht the n derivtive f n) :, b) R stisfies the condition m f n) M for ll x, b), we hve the inequlity n )n f t) dt + ) n ) j 33) j! k j { hi ) j ) j δ i f j ) hi x i+) + δ i f j ) x i )} f n ) b) f n ) ) b ) n + )! n+ ) k ) n+ hi ) ) r n + δi { + ) r } r r0 h i

85 77 A sofo where M m n+ r0 b k n + ) n!) ) n+ hi ) ) r n + δi { + ) } r r h i k ) n+ n+ hi ) ) ) r n + δi { + ) r } n + )! r r0 h i : x i+ x i nd δ i : α i+ x i+ + x i, i 0,, k h i Proof We utilise 3) nd 3), multiply through by b ) nd choose h t) : K n,k t) s defined by 3) nd g t) : f n) t), t, b such tht K n,k t) f n) t) dt f n) 34) t) dt K n,k t) dt b Γ γ b ) b b ) Kn,k t) dt K n,k t) dt Now we my evlute nd G : n + )! f n) t) dt f n ) b) f n ) ) k xi+ K n,k t) dt x i n! t α i+) n dt k Using the definitions of h i nd δ i we hve { x i+ α i+ ) n+ + α i+ x i ) n+} nd such tht G n + )! x i+ α i+ h i δ i α i+ x i h i + δ i { k hi δ i ) n+ ) } n+ hi + + δ i

86 INTEGRAL INEQUALITIES FOR n TIMES DIFFERENTIABLE MAPPINGS 78 n + )! k n+ n+ n + + r r0 n + )! k n + r r0 ) δ r i hi ) ) n+ r δ i ) r hi ) n+ r hi ) n+ n+ ) ) r n + δi { + ) } r r r0 h i Also, G : k Kn,k t) dt n + ) n!) xi+ x i n t α i+) n!) dt k {x i+ α i+ ) n+ + α i+ x i ) n+} { k hi ) n+ ) } n+ n + ) n!) δ hi i + + δ i n + ) n!) k hi From identity ), we my write k ) n+ n+ r0 K n,k t) f n) t) dt ) n f t) dt + ) n ) ) r n + δi { + ) } r r h i n ) j j! j { } x i+ α i+ ) j f j ) x i+ ) α i+ x i ) j f j ) x i ) nd from the left hnd side of 34) we obtin G 3 : ) n f t) dt + ) n ) j j! j k { hi ) j ) j δ i f j ) hi x i+) + δ i f j ) x i )} n f n ) b) f n ) ) k ) hi b ) n + )! n+ ) ) r n + δi { + ) } r r r0 fter substituting for G h i ) n+

87 79 A sofo From the right hnd side of 34) we substitute for G nd G b ) b G 4 : b ) Kn,k t) dt K n,k t) dt Hence, k b n + ) n!) k n + )! hi hi ) n+ n+ r0 so tht ) ) r n + δi { + ) } r r ) n+ n+ ) ) ) r n + δi { + ) r } r r0 G 3 M m G 4 ) nd Theorem hs been proved Corollry Let f, I k nd α k be defined s in Theorem nd further define 35) δ α i+ x i+ + x i for ll i 0,, k such tht 36) δ min {h i i,, k} The following inequlity pplies 37) )n f t) dt + ) n k { i) j hi δ i f n ) b) f n ) ) b ) n + )! M m b n + ) n!) k n+ n + r r0 k n+ n + n + )! r r0 n j j! ) j f j ) hi x i+) ) k n+ r0 δ r hi h i + δ i h i ) ) n+ r δ r hi { + ) } r ) j f j ) x i )} ) n+ r { + ) } r ) ) n+ r δ r hi { + ) r } ) The proof follows directly from 33) upon the substitution of 35) nd some minor simplifiction

88 INTEGRAL INEQUALITIES FOR n TIMES DIFFERENTIABLE MAPPINGS 80 Remrk If for ny division I k : x 0 < x < < x k < x k b of the intervl, b, we choose δ 0 in 35), we hve the inequlity n )n f t) dt + ) n 38) j! j k ) j hi { ) j f j ) x i+) f j ) x i )} f n ) b) f n ) ) b ) n + )! M m b ) k n + ) n!) ) k ) n+ hi ) n+ hi n + )! k hi ) n+ ) The proof follows directly from 37) Remrk Let f n) be defined s in Theorem nd consider n equidistnt prtitioning E k of the intervl, b, where ) b E k : x i + i, i 0,, k k The following inequlity pplies 39) )n f t) dt + ) n n j j! b { ) )} ) j k i ) + b i ) k i) + ib f j ) f j ) k k k ) j f n ) b) f n ) ) ) n+ ) b b ) n + )! k M m) nk n + )! n + ) n+ b k Proof The proof follows upon noting tht h i x i+ x i ) b k, i 0,, k 6 Some Prticulr Integrl Inequlities In this subsection we point out some specil cses of the integrl inequlities in Section 3 In doing so, we shll recover, subsume nd extend the results of number of previous published ppers 3, 4, 5

89 8 A sofo We shll recover the left nd right rectngle inequlities, the perturbed trpezoid inequlity, the midpoint nd Simpson s inequlities nd the Newton-Cotes three eighths inequlity, nd Boole type inequlity In the cse when n, for the kernel K,k t) of 4), the inequlities 3), 6) nd 8) reduce to the results obtined by Drgomir 9, 0, for the cses when f :, b R is bsolutely continuous nd f belongs, respectively to the L, b, L p, b nd L, b spces Similrly, for n, Drgomir 7 extended Theorem 7 for the cse when f, b R is function of bounded vrition on, b For n, nd the kernel K,k t) of 4), the following theorem is obtined Theorem 3 Let I k : x 0 < x < < x k < x k b be division of the intervl, b, α i x i, x i i,, k) nd α k+ b If f :, b R is bsolutely continuous, then we hve the inequlity 40) + f t) dt k k α i+ α i ) f x i ) { x i+ α i+ ) f x i+ ) x i α i+ ) f x i )} f 6 f p k q + ) q α i+ x i ) 3 + x i+ α i+ ) 3 if f L, b, k α i+ x i ) q+ + x i+ α i+ ) q+) q if f L p, b, p > nd p + q, ) ν h) 6 + ρ δ) 4 + νh) mx i i 0,k f 3 8 f ν h) if f L, b, where h i : x i+ x i, ν h) mx {h i i 0,, k }, nd ρ δ) mx {δ i i 0,, k } δ i α i+ x i+ + x i Proof The first nd second prt of the inequlity 40) cn be obtined directly from Theorem 7 nd Theorem 8 respectively We now prove the third line of the inequlity 40), on the L, b spce, which is n improvement of the inequlity 8) for the cse n

90 INTEGRAL INEQUALITIES FOR n TIMES DIFFERENTIABLE MAPPINGS 8 From 4) K,k t) f t) dt Now k K,k t) f t) dt k xi+ xi+ x i k K,k t) f t) dt x i k xi+ x i sup t x i,x i+ mx i 0,k Now, we my observe tht { α i+ x i ) mx i 0,k K,k t) f t) dt xi+ x i t α i+ ) f t) dt : W t α i+ ) f t) dt t α i+ ) xi+ f t) dt x i { } α i+ x i ), x i+ α i+ ) xi+ f t) dt }, x i+ α i+ ) { x i+ α i+ ) + α i+ x i ) mx i 0,k Using the identity reduces the previous line to A ) ) ) + B A B A + B + { x i+ x i ) mx + i 0,k 8 + x i+ x i ) α i+ x i+ + x i ) h mx i δ + mx i i 0,k 6 i 0,k 4 x i } α i+ x i ) x i+ α i+ ) + α i+ x i+ + x i } ) ) ) hi + mx i 0,k δ i Hence, from ν h) 6 + ρ δ) 4 ν h) W 6 + ν h) + ρ δ) 4 mx δ i i 0,k + ν h) ) mx δ i f i 0,k

91 83 A sofo nd the third line of the inequlity 40) is thus proved We my lso note tht δ i α i+ x i+ + x i h i, δ i hi ), nd hence ν h) W 6 + ν h) 6 ) + ν h) f ν h) f nd Theorem 3 is completely proved For the two brnch Peno kernel, 4) K n, t) : n! t )n, t, x) n! t b)n, t x, b the following results were obtined by Cerone, Drgomir nd Roumeliotis 6 which follow directly from Theorems 7, 8 nd 9 Theorem 4 Let f :, b R be mpping such tht the derivtives f n ) n ) re bsolutely continuous on, b nd x, b Then n 4) ) j { f t) dt + x ) j ) j b x) j} f j ) x) j! j f { x ) n+ + b x) n+} if f n) L, b, n + )! f p n! nq + ) q f n) b n! x ) nq+ + b x) nq+) q + x + b f n) n! ) n if f n) L p, b, p > nd p + q, b ) n if f n) L, b, Proof The proof follows from Theorems 7, 8 nd 9, nd using the kernel 4) fter substituting α 0 α x 0, α α 3 x b nd x x, b

92 INTEGRAL INEQUALITIES FOR n TIMES DIFFERENTIABLE MAPPINGS 84 From Theorem 9 we note tht K n, t) f n) t) dt f n) K n, t) f n) sup K n, t) t,b f n) mx {x, b x}) n n! f n) b + n! x + b n f ) n) b ) n, n! which completes the proof of Theorem 4 Corollry 5 If in 4) we substitute x b, we obtin the perturbed right rectngle inequlity 43) f t) dt + n ) j b ) j f j ) b) j! j f n + )! b )n+ if f n) L, b, f p n! b )n+ q nq + ) q if f n) L p, b, p > nd p + q, f n) b ) n if f n) L, b n! If in 4) we substitute x, we obtin the perturbed left rectngle inequlity 44) f t) dt j n b ) j f j ) ) j! f n + )! b )n+ if f n) L, b, f p n! b )n+ q nq + ) q if f n) L p, b, p > nd p + q, f n) b ) n if f n) L, b n!

93 85 A sofo If in 4) we substitute x +b, we obtin the best estimte, perturbed trpezoid inequlity n ) j b { } ) f t) dt ) j + b 45) f j ) j! j f n + )! n b )n+ if f n) L, b, f p b ) n+ q n! nq + ) q n if f n) L p, b, f n) b ) n p > nd p + q, n! n if f n) L, b Remrk 3 It is of interest to note tht only the even derivtives occur in the left hnd side of 45) Tking n in Theorem 4 reproduces some of the results obtined by Drgomir nd Wng 9, 0, nd for n we recover the results obtined by Cerone, Drgomir nd Roumeliotis 3, 4, 5 Moreover, we my observe tht the bounds given in 4) for generlised interior point method obtined from investigting vrious norms of the kernel 4) re the sme s the bounds obtined from the generlised trpezoidl type rule resulting from vrious norms of the Peno kernel given by x t)n n! The following corollry my be obtined from 43) nd 44) Corollry 6 Let f be defined s given in Theorem 4 Then 46) f t) dt + n b ) j { )} ) j f j ) b) f j ) j! j f n) n + )! f n) p n! b )n+ b )n+ q nq + ) q if n r r+ r if n r + if f n) L, b, if f n) L p, b, p > nd p + q, f n) b ) n if f n) L, b n! Proof Using the identity 6) with the kernel 4) t the left nd right corners, we obtin 46) Detils my lso be seen in the pper by Cerone, Drgomir nd Roumeliotis 3 for the norm

94 INTEGRAL INEQUALITIES FOR n TIMES DIFFERENTIABLE MAPPINGS 86 Corollry 7 Let f be defined s bove Then n ) j ) 47) f t) dt + b α) j f j ) b) ) j α ) j f j ) j! j f α ) n+ + b α) n+), n + )! f p n! nq + ) q f n) b n! α ) nq+ + b α) nq+) q, + α + b n ) Proof Follows from Theorem 7 with α 0, x 0, x b, α b nd α α, b The following inequlities relte to Tylor like expnsions Corollry 8 Let g be defined s in Corollry Then we hve the inequlity n 48) ) j { g y) g ) + x ) j ) j y x) j} g j) x) j! j g n+) { x ) n+ + y x) n+} if g n+) L, y n + )! g n+) p n! nq + ) q x ) nq+ + y x) nq+) q if g n+) L p, y, p > nd p + q, g n+) y + n! x + y n if g n+) L, y for ll x y Proof Follows directly from 4) by choosing b y nd f g Perturbed right nd left inequlities my lso be obtined from 43) nd 44) respectively by putting x nd x y It is lso well known tht for the clssicl Tylor expnsion round some point we hve the following inequlity, n 49) g y) y ) j n+ g j) ) y ) j! g n+) n + )! for ll y j0

95 87 A sofo We therefore my see tht the pproximtion 48) round the rbitrry point x, y provides better pproximtion for the mpping g t point y thn the clssicl Tylor expnsion 49) round point The inequlity 48) ttins its optimum when x +b in 48), so tht we hve the inequlity n ) j y { } ) g y) g ) + ) j + y 50) g j) j! j g n+) y )n+ n + )! if g n+) L n, y g n+) p y )n+ q n if g n+) L p, y, n! nq + ) q p > nd p + q, g n+) ) n y if g n+) L, y n! The inequlity 50) shows tht for g C, y the series ) j y { } ) g ) ) j + y g j) j! j converges more rpidly to g y) thn the usul one, n ) j y g j) ) from 49) j0 If we choose n in 4) we obtin the Ostrowski inequlity b ) f t) dt b ) f x) x +b 4 + b ) b ) f lso the midpoint inequlity ) + b ) b f t) dt b ) f f nd the trpezoid inequlity f t) dt b f ) + f b)) b ) f For n in 4) we deduce the following results, lso obtined by Cerone, Drgomir nd Roumeliotis 3 f t) dt b ) f x) + b ) x + b ) f x) 4 + x +b b ) ) b ) 3 f

96 INTEGRAL INEQUALITIES FOR n TIMES DIFFERENTIABLE MAPPINGS 88 The clssicl midpoint inequlity, ) + b f t) dt b ) f 4 b )3 f, nd the perturbed trpezoid inequlity f t) dt b ) b f ) + f b)) f b) f )) b )3 6 f From 48) we obtin g y) g ) + y ) g x) + y ) x + y ) g x) g y ) x +y y ) ) ) if g L, y, g p x ) q+ + y x) q+) q if g L 6 q + ) p, y, q p > nd p + q, g y + x + y ) if g L, y The following theorem produces nother integrl inequlity with mny pplictions Theorem 9 Let f :, b R be n bsolutely continuous mpping on, b nd x b, α x α b Then we hve 5) n ) j f t) dt + α ) j f j ) ) j! j { + x α ) j x α ) j} f j ) x ) + b α ) j f j ) b)

97 89 A sofo f n) α ) n+ + x α ) n+ + α x ) n+ + b α ) n+) n + )! f n) x ) n+ + b x ) n+) n + )! f n) n + )! b )n+, f n) L, b, f n) p α ) nq+ + x α ) nq+ + α x ) nq+ + b α ) nq+) q, n! nq + ) q f n) L p, b, p > nd p + q f n) b ) n, f n) L, b n! Proof Consider the division x 0 x x b nd the numbers α 0, α, x ), α x, b nd α 3 b From the left hnd side of 3) we obtin n ) j j j! n ) j j j! { x i α i ) j x i α i+ ) j} f j ) x i ) α ) j f j ) ) + { x α ) j x α ) j} f j ) x ) + b α ) j f j ) b) From the right hnd side of 3) we obtin f n) {α i+ x i ) n+ + x i+ α i+ ) n+} n + )! f n) α ) n+ + x α ) n+ + α x ) n+ + b α ) n+) n + )! nd hence the first line of the inequlity 5) follows The p norm nd norm inequlities follow from 6) nd 8) respectively Hence Theorem 9 is proved Notice tht if we choose α nd α b in Theorem 9 we obtin the inequlity 4) of Theorem 4 The following proposition embodies number of results, including the Ostrowski inequlity, the midpoint nd Simpson s inequlities nd the three-eighths Newton- Cotes inequlity including its generlistion

98 INTEGRAL INEQUALITIES FOR n TIMES DIFFERENTIABLE MAPPINGS 90 Proposition 30 Let f be defined s in Theorem 9 nd let x b, nd m )+b m x +m )b m b for m nturl number, m, then we hve the inequlity 5) P m,n b n b : ) j ) j { f t) dt + f j ) b) ) j f )} j ) j! m j { + x ) b ) j ) } j b m m b x ) f j ) x ) f n) ) n+ )) n+ b b + x n + )! m m )) ) n+ b + b x m if f n) L, b, f n) p b n! nq + ) q m b + b x m ) nq+ + x )) nq+ ) q, if f n) L p, b, p > nd p + q f n) b ) n, if f n) L, b n! b m )) nq+ Proof From Theorem 9 we note tht m ) + b + m ) b α nd α m m so tht ) b α, b α b m m, ) ) b b x α x nd x α b x ) m m From the left hnd side of 5) we hve { α ) j f j ) ) + x α ) j x α ) j} f j ) x ) + b α ) j f j ) b) ) j b { f j ) b) f )} j ) m { )) j b + x m ) ) } j b b x ) f j ) x ) m

99 9 A sofo From the right hnd side of 5), α ) n+ + x α ) n+ + α x ) n+ + b α ) n+ ) n+ )) n+ b b b + x + b x m m m )) n+ nd the first line of the inequlity 5) follows The second nd third lines of the inequlity 5) follow directly from 5), hence the proof is complete The following corollry points out tht the optimum of Proposition 30 occurs t x α+α +b in which cse we hve: Corollry 3 Let f be defined s in Proposition 30 nd let x +b in which cse we hve the inequlity ) + b 53) P m,n : f t) dt + n b ) j ) j { f j ) b) ) j f )} j ) j! m j ) j m ) b ) + ) j) ) + b f j ) m f n) ) n+ ) ) n+ b m + n + )! m if f n) L, b, f n) p n! ) nq + q b m ) n+ q + if f n) L p, b, p > nd p + q f n) b ) n, if f n) L, b n! m ) nq+ ) q The proof follows directly from 5) upon substituting x +b A number of other corollries follow nturlly from Proposition 30 nd Corollry 3 nd will now be investigted

100 INTEGRAL INEQUALITIES FOR n TIMES DIFFERENTIABLE MAPPINGS 9 Corollry 3 Let f be defined s in Proposition 30 Then 54) lim P m,n m n ) j { f t) dt + x ) j x b) j} f j ) x ) j! j f n) x ) n+ + b x ) n+) n + )! f n) p n! nq + ) q f n) b n! x ) nq+ + b x ) nq+) q + x + b nd this is the result obtined in Theorem 4 In this cse we see tht α nd α b nd the optimum of 53) occurs when x +b in which cse we recover the result 45) The following two corollries generlise the Simpson inequlity nd follow directly from 5) nd 53) for m 6 Corollry 33 Let the conditions of Corollry 3 hold nd put m 6 Then we hve the inequlity 55) P 6,n b n b : ) j ) j { f t) dt + f j ) b) ) j f )} j ) j! 6 j { + x 5 + b ) j x + 5b ) } j f j ) x ) 6 6 f n) ) n+ b + x 5 + b ) n+ n + )! x + + 5b ) ) n+, f n) L, b, 6 f n) p b n! nq + ) q 6 + x + + 5b 6 ) n ) nq+ + x 5 + b 6 ) n+ ) q, ) nq+ if f n) L p, b, p > nd p + q f n) b ) n, if f n) L, b, n!

101 93 A sofo which is the generlised Simpson inequlity Corollry 34 Let the conditions of Corollry 3 hold nd put m 6 Then t the midpoint x +b we hve the inequlity ) + b 56) P 6,n b n b : ) j ) j { f t) dt + f j ) b) ) j f )} j ) j! 6 j ) j b + ) j) ) + b f j ) 3 f n) n + )! f n) p n! ) n+ b ) + n+, f n) L, b, 6 ) nq + q b 6 ) n+ q + nq+ ) q, if f n) L p, b, p > nd p + q f n) b ) n, if f n) L, b n! Remrk 4 Choosing n in 56) we hve ) 57) + b ) { ) P b + b 6, : f t) dt f ) + 4f + f b)} 6 58) 5 36 f b ), f L, b, ) ) f q + q b ) p + q+ q, q + 6 if f L p, b, p > nd p + q f b ), if f n) L, b Remrk 5 Choosing n in 40) we hve perturbed Simpson type inequlity ) + b 59) P 6, ) ) ) b + b : f t) dt f ) + 4f + f b) 6 ) b f b) f ) ) + 6

102 INTEGRAL INEQUALITIES FOR n TIMES DIFFERENTIABLE MAPPINGS 94 b ) 3 f 7, f L, b, ) + ) b q q ) + q+ q 6 q + f p, f L p, b, p > nd p + q b ) f, f L, b Corollry 35 Let f be defined s in Corollry 3 nd let m 4, then we hve the inequlity ) + b 60) P 4,n n : ) j ) j b f t) dt + j! 4 j ) f j ) b) ) j f j ) ) + ) j) ) + b f j ) f n) n + )!4 n b )n+, f n) L, b, f n) ) ) p 4 q n+ q b, f n) L p, b, p > nd n! nq + 4 p + q f n) b ) n, f n) L, b n! Remrk 6 From 60) we choose n nd we hve the inequlity ) + b 6) P 4, ) )) b + b : f t) dt f b) + f ) + f 4 ) b f b) f ) ) + 4 f b ) 3, f L, b, 96 f ) ) p 4 q + q b, f L p, b, p > nd q + 4 p + q b ) f, f L, b Theorem 36 Let f :, b R be n bsolutely continuous mpping on, b nd let < x x b nd α, x ), α x, x ) nd α 3 x, b Then we

103 95 A sofo hve the inequlity n ) j 6) f t) dt + α ) j f j ) ) j! j + x α ) j x α ) j) f j ) x ) + x α ) j x α 3 ) j) f j ) x ) + b α 3 ) j f b) j ) f n) α ) n+ + x α ) n+ + α x ) n+ n + )! + x α ) n+ + α 3 x ) n+ + b α 3 ) n+) f n) L, b, f n) p n! nq + ) q α ) nq+ + x α ) nq+ + α x ) nq+ + x α ) nq+ + α 3 x ) nq+ + b α 3 ) nq+) q, f n) L p, b, p > nd p + q f n) b ) n, f n) L, b n! Proof Consider the division x 0 < x < x b, α, x ), α x, x ), α 3 x, b, α 0, x 0, x 3 b nd put α 4 b From the left hnd side of 3) we obtin n ) j j j! 3 { x i α i ) j x i α i+ ) j} f j ) x i ) n ) j α ) j f j ) ) j! j + x α ) j x α ) j) f j ) x ) + x α ) j x α 3 ) j) f j ) x ) + b α 3 ) j f j ) b) From the right hnd side of 3) we obtin f n) {α i+ x i ) n+ + x i+ α i+ ) n+} n + )! f n) α ) n+ + x α ) n+ + α x ) n+ n + )! + x α ) n+ + α 3 x ) n+ + b α 3 ) n+) nd hence the first line of the inequlity 6) follows The p norm nd norm inequlities follow from 6) nd 8) respectively, hence Theorem 36 is proved

104 INTEGRAL INEQUALITIES FOR n TIMES DIFFERENTIABLE MAPPINGS 96 Corollry 37 Let f be defined s in Theorem 36 nd consider the division m ) + b + m ) b α α α 3 b m m for m nturl number, m Then we hve the inequlity n Q m,n : ) j 63) f t) dt + α ) j f j ) ) j! j { m ) j ) } j ) + b m ) + b + α α f j ) x ) m m + { + m ) b m + b α 3 ) j f j ) b) f n) n + )! R m,n : + α α 3 ) j ) } j + m ) b α α 3 f j ) x ) m f n) α ) n+ m ) + b + n + )! m ) n+ m ) + b + m ) b + α ) n+ ) n+ α m m ) ) n+ + m ) b + + b α 3 ) n+, f n) L, b, m f n) p R n! nq + ) m,nq ) q, f n) L p, b, p > nd p + q, q f n) b ) n, f n) L, b n! Proof Choose in Theorem 36, x m )+b m, nd x +m )b m, hence the theorem is proved Remrk 7 For prticulr choices of the prmeters m nd n, Corollry 37 contins generlistion of the three-eighths rule of Newton nd Cotes The following corollry is consequence of Corollry 37 Corollry 38 Let f be defined s in Theorem 36 nd choose α +b x +x, then we hve the inequlity b n Qm,n : ) j 64) f t) dt + α ) j f j ) ) j! j { ) } j + x α ) j ) j m ) b ) f j ) x ) m { m ) } j ) b ) + x α 3 ) j f j ) x ) m + b α 3 ) j f j ) b)

105 97 A sofo f n) n + )! R m,n : + f n) α ) n+ m ) + b + n + )! m b ) m ) m ) n+ + α 3 + m ) b m α ) n+ ) n+ + b α 3 ) n+), f n) L, b, f n) p ) q Rm,nq, f n) L p, b, p > nd p + q, n! nq + ) q f n) b ) n, f n) L, b n! Proof If we put α +b 64) nd the corollry is proved x+x into 63), we obtin the inequlity The following corollry contins n optimum estimte for the inequlity 64) Corollry 39 Let f be defined s in Theorem 36 nd mke the choices ) ) 3m 4 4 m α + b nd m m ) ) 4 m 3m 4 α 3 + b m m then we hve the best estimte ˆQ 65) m,n n : ) j f t) dt + j! j b ) j ) 4 m) { f j ) b) ) j f )} j ) m ) j b ) m ) { + ) j} ) f j ) x ) + f j ) x ) m f n) ) n+ b 4 m) n+ + m ) n+) n + )! m f n) L, b, f n) ) n+ p b q 4 m) nq+ + m ) nq+) q, n! nq + ) q m f n) L p, b, p > nd p + q, f n) b ) n, f n) L, b n!

106 INTEGRAL INEQUALITIES FOR n TIMES DIFFERENTIABLE MAPPINGS 98 Proof Using the choice α we my clculte +m )b m nd α ) +b x+x, x m )+b m nd x 4 m) b ) m b α 3 ) x α ) α x ) x α ) α 3 x ) m ) b ) m Substituting in the inequlity 64) we obtin the proof of 65) Remrk 8 For m 3, we hve the best estimtion of 65) such tht ˆQ b n 3,n : ) j 66) f t) dt + j! j b ) j { ) j f j ) b) ) j b f )} j ) { ) j} ) )) + b + b f j ) + f j ) f n) 6 n n + )! b )n+, f n) L, b, f n) ) ) n+ p q 3 b q n! nq+, 6 f n) L p, b, p > nd p + q, f n) b ) n, f n) L, b n! then Proof From the right hnd side of 65), consider the mpping ) n+ ) n+ 4 m m M m,n : + m m M m,n n + ) m nd M m,n ttins its optimum when m ) n + ) n ) m m m, in which cse m 3 Substituting m 3 into 65), we obtin 66) nd the corollry is proved

107 99 A sofo When n, then from 66) we hve ˆQ 3, : b f t) dt 6 ) f b) + f )) + ) b f b) f )) 6 ) ) )) b + b + b + f + f 3 f 6 b ) 3, f L, b, ) ) 3 f q + q b p, q + 6 f L p, b, p > nd p + q, f b ), f L, b The next corollry encpsultes the generlised Newton-Cotes inequlity Corollry 40 Let f be defined s in Theorem 36 nd choose x + b, x + b, α + b 3 3, r ) + b + r ) b α nd α 3 r r Then for r nturl number, r 3, we hve the inequlity 67) T r,n b n b : ) j ) j { f t) dt + f j ) b) ) j f )} j ) j! r j { b ) j ) } j ) r 3) + ) j b f j ) x ) 3r 6 { b ) j ) } j + ) j b ) r 3) f j ) x ) 6 3r

108 INTEGRAL INEQUALITIES FOR n TIMES DIFFERENTIABLE MAPPINGS 00 f n) n + )! f n) L, b, b ) n+ r 3 r n+ + 3r ) ) n+ + 6 n+, f n) ) n+ p b q ) 6 nq+ + r r 3)) nq+ + r nq+ q, n! nq + ) q 6r f n) L p, b, p > nd p + q f n) b ) n, f n) L, b n! Proof From Theorem 36, we put x +b 3, x +b 3, α +b, α nd α 3 +r )b r Then 67) follows r )+b r Remrk 9 The optimum estimte of the inequlity 67) occurs when r 6 from 67) consider the mpping M r,n : r n+ + r 3 3r ) n+ + 6 n+ the M r,n n + ) r n + ) n+ r 3 ) n r nd Mr,n ttins its optimum when r 3 r, in which cse r 6 In this cse, we obtin the inequlity 66) nd specificlly for n, we obtin Simpson type inequlity 68) ˆQ 3, ) b : f t) dt f ) + f b)) 6 ) ) )) b + b + b f + f f b ) f L, b, ) 3 f p q + q b 6 ) + q, f L p, b, p > nd p + q f b ), f L, b Corollry 4 Let f be defined s in Theorem 36 nd choose m 8 such tht α 7+b 8, α +b 8 nd α 3 +7b 8 with x +b 3 nd x +b 3 Then we

109 0 A sofo hve the inequlity 69) T 8,n b n b : ) j ) j f t) dt + f j ) b) ) j f )) j ) j! 8 j ) j 5 ) ) j b ) + ) j f j ) x ) ) j f j ) x ) 6 4 f n) S n : f n) ) n+ b 3 n+ + 4 n+ + 5 n+), n + )! n + )! 4 f n) L, b, f n) p S n! nq + ) nq ) q, q f n) L p, b, p > nd p + q f n) b ) n, f n) L, b n! where ) n+ b S n : 3 n+ + 4 n+ + 5 n+) 4 nd ) nq+ b S nq : 3 nq+ + 4 nq+ + 5 nq+) 4 Proof From Theorem 36 we put x + b 3, x + b 3 nd α +b nd the inequlity 69) is obtined, α 7 + b, α 3 + 7b 8 8 When n we obtin from 69) the three-eighths rule of Newton-Cotes

110 INTEGRAL INEQUALITIES FOR n TIMES DIFFERENTIABLE MAPPINGS 0 Remrk 0 From 67) with r 3 we hve T 3,n : b n b ) j ) j f t) dt + f j ) b) ) j f )) j ) j! 3 j ) j b { } + f j ) x ) f j ) x ) 6 f n) b ) n+ n + )! 3 n+ + f n) L, b, 6 n+ ), f n) ) n+ p b q ) + nq+ q, n! nq + ) q 6 f n) L p, b, p > nd p + q f n) b ) n, f n) L, b n! In prticulr, for n, we hve the inequlity T 3, : b f t) dt 3 ) b + b f 6 3 b + 6 f 7 ) f +b 3 ) f b) + f )) + ) + f ) f +b 3 )) + b 3 )) b ) 3, f L, b, ) + b f p 6 f q + q+ q + ) b f b) f ) ) 3 ) q, f L p, b, p > nd p + q b ), f L, b The following theorem encpsultes Boole s rule Theorem 4 Let f :, b R be n bsolutely continuous mpping on, b nd let < x < x < x 3 < b nd α, x ), α x, x ), α 3 x, x 3 ) nd

111 03 A sofo α 4 x 3, b Then we hve the inequlity 70) n ) j f t) dt + α ) j f j ) ) j! j + x α ) j x α ) j) f j ) x ) + x α ) j x α 3 ) j) f j ) x ) + x 3 α 3 ) j x 3 α 4 ) j) f j ) x 3 ) + b α 4 ) j f b) j ) f n) n + )! α ) n+ + x α ) n+ + α x ) n+ + x α ) n+ + α 3 x ) n+ + x 3 α 3 ) n+ + α 4 x 3 ) n+ + b α 4 ) n+) if f n) L, b, f n) p α n! nq + ) ) nq+ + x α ) nq+ + α x ) nq+ q + x α ) nq+ + α 3 x ) nq+ + x 3 α 3 ) nq+ + α 4 x 3 ) nq+ + b α 4 ) nq+ q, if f n) L p, b, p > nd p + q, f n) b ) n, f n) L, b n! Proof Follows directly from 3) with the points α 0 x 0, x 4 α 5 b nd the division x 0 < x < x < x 3 b, α, x ), α x, x ), α 3 x, x 3 ) nd α 4 x 3, b The following inequlity rises from Theorem 4 Corollry 43 Let f be defined s in Theorem 36 nd choose α +b α +7b 8, α 3 7+b 8, α 4 +b, x 7+b 9, x 3 +7b 9 nd x x+x3, then we cn stte: +b, b n b ) j ) j { f t) dt + f j ) b) ) j f )} j ) j! j ) { j 5 ) } j { ) )} b + ) j 7 + b f j ) ) j + 7b f j ) b 9 ) j { ) j} f j ) + b )

112 INTEGRAL INEQUALITIES FOR n TIMES DIFFERENTIABLE MAPPINGS 04 f n) ) n+ b 3 n+ + 4 n+ + 5 n+ + 6 n+), n + )! 36 if f n) L, b, f n) ) n+ p b q 3 nq+ + 4 nq+ + 5 nq+ + 6 nq+) q, n! nq + ) q 36 if f n) L p, b, p > nd p + q f n) b ) n, if f n) L, b n! 7 Applictions for Numericl Integrtion In this section we utilise the prticulr inequlities of the previous sections nd pply them to numericl integrtion Consider the prtitioning of the intervl, b given by m : x 0 < x < < x m < x m b, put h i : x i+ x i i 0,, m ) nd put ν h) : mx h i i 0,, m ) The following theorem holds Theorem 44 Let f :, b R be bsolutely continuous on, b, k nd m Then we hve the composite qudrture formul 7) where f t) dt A k m, f) + R k m, f) 7) A k m, f) : T k m, f) U k m, f), 73) T k m, f) : nd m n j ) j hi f j ) x i ) + ) j f j ) x i+ ) k j! 74) U k m, f) : m n j k r ) j hi k j! { } ) ) j k r) f j ) xi + rx i+ k is perturbed qudrture formul The reminder R k m, f) stisfies the estimtion 75) R k m, f)

113 05 A sofo f n) m n n + )!k n+ h n+ i, if f n) L, b, f n) p n! k nq + )) q k) n m h nq+ i ) q, if f n) L p, b, p > nd p + q, f n) m f n) n!k n h n i n!k n νn h), if f n) L, b where ν h) : mx h i i 0,, m ) Proof We shll pply Corollry on the intervl x i, x i+, i 0,, m ) Thus we obtin xi+ f t) dt + x i r n j ) j hi f j ) x i ) + ) j f j ) x i+ ) k j! k { } ) + ) j k r) f j ) xi + rx i+ k n + )! n sup f n) t) ) n+ xi+ x i, t x i,x i+ k n! k) n k nq + )) q x i+ x i ) n+ q xi+ x i f n) t) ) p p dt, xi+ n!k n x i+ x i ) n f n) t) dt x i

114 INTEGRAL INEQUALITIES FOR n TIMES DIFFERENTIABLE MAPPINGS 06 Summing over i from 0 to m nd using the generlised tringle inequlity, we hve m xi+ n ) j hi f t) dt + f j ) x i ) + ) j f j ) x i+ ) x i k j! r j k { } ) + ) j k r) f j ) xi + rx i+ k m n + )! n h n+ i k n+ sup f n) t), t x i,x i+ n! k) n k nq + )) q m h n+ q i xi+ x i f n) t) ) p p dt, Now, 76) m n!k n h n i xi+ m f t) dt + x i n j f n) t) dt ) j hi f j ) x i ) + ) j f j ) x i+ ) k j! As + m n j R k m, f) sup t x i,x i+ ) j hi k j! k r { } ) ) j k r) f j ) xi + rx i+ k f n) t) f n), the first inequlity in 75) follows Using the discrete Hölder inequlity, we hve, from m n! k) n h n+ q k nq + )) i q x i+ x i f n) t) p dt p n! k) n k nq + )) q m x i+ x i n! k nq + )) q k) n m f n) t) p dt m ) ) q h n+ q q i p p h nq+ i p ) q f n) p

115 07 A sofo nd the second line of 75) follows Now, we my observe tht x m i+ n!k n h n i f n) t) dt m n!k n mx h i) n,,m ) x i n!k n ν h))n f n) Hence, from 76) we see tht f t) dt + T k m, f) + U k m, f) R k m, f) nd the theorem is proved x i+ x i f n) t) dt The following corollry holds Corollry 45 Let f be defined s bove, then we hve the equlity 77) where T m, f) : f t) dt T m, f) U m, f) + R m, f) m n j ) j hi f j ) x i ) + ) j f j ) x i+ ) 4 j! U m, f) is the perturbed midpoint qudrture rule, contining only even derivtives m n ) j hi { } ) U m, f) : ) j f j ) xi + x i+ 4 j! j nd the reminder, R m, f) stisfies the estimtion R m, f) f n) m n+ n + )! f n) p h n+ i, if f n) L, b m h nq+ i ) q, if f n) L p, b, n! nq + )) q n p > nd p + q, f n) n! n νn h), if f n) L, b Corollry 46 Let f nd m be defined s bove Then we hve the equlity 78) f t) dt T 3 m, f) U 3 m, f) + R 3 m, f),

116 INTEGRAL INEQUALITIES FOR n TIMES DIFFERENTIABLE MAPPINGS 08 where nd T 3 m, f) : m n j U 3 m, f) : m ) j hi f j ) x i ) + ) j f j ) x i+ ) 6 j! n j ) ) j j! f j ) xi + x i+ 3 ) j hi 6 ) ) + f j ) xi + x i+ 3 nd the reminder stisfies the bound R 3 m, f) f n) m 3 6 n n + )! f n) p n! 3 nq + )) q 6 n h n+ i, if f n) L, b m h nq+ i ) q, if f n) L p, b, p > nd p + q, f n) m n!3 n h n i, if f n) L, b Theorem 47 Let f nd m be defined s bove nd suppose tht ξ i x i, x i+ i 0,, m ) Then we hve the qudrture formul: 79) f t) dt m n ) j { ξ j! i x i ) j f j ) x i ) j } ) j x i+ ξ i ) j f j ) x i+ ) + R ξ, m, f) nd the reminder, R ξ, m, f) stisfies the inequlity R ξ, m, f) f n) n + )! f n) p m n! nq + ) q ξ i x i ) n+ + x i+ ξ i ) n+), m if f n) L, b { ξ i x i ) nq+ + x i+ ξ i ) nq+}) q, if f n) L p, b, p > nd p + q, f n) m h n i n!, if f n) L, b

117 09 A sofo Proof From Theorems 7, 8 nd 9 we put α 0, x 0, x b, α b nd α α, b such tht f t) dt + R ξ, m, f) n ) j j j! α) j f j ) ) + b α) j f j ) b) Over the intervl x i, x i+ i 0,, m ), we hve x i+ n ) j f t) dt + j! x j i ) j ξ i x i ) j f j ) x i ) R ξ, m, f) x i+ ξ i ) j f j ) x i+ ) nd therefore, using the generlised tringle inequlity R ξ, m, f) m x i+ f t) dt + x i m j n ) j j! x i+ ξ i ) j f j ) x i+ ) ) j ξ i x i ) j f j ) x i ) m n + )! sup t x i,x i+ f n) t) ξ i x i ) n+ + x i+ ξ i ) n+), n! nq + ) q m xi+ x i f n) t) ) p p dt { ξ i x i ) nq+ + x i+ ξ i ) nq+} q, xi+ n! x i+ x i ) n x i f n) t) dt The first prt of the inequlity 80) follows, since we hve sup t x i,x i+ f n) t) f n)

118 INTEGRAL INEQUALITIES FOR n TIMES DIFFERENTIABLE MAPPINGS 0 In the cse tht f n) L p, b we utilise the discrete Hölder inequlity nd for the second prt of 80) we hve n! nq + ) q m { ξ i x i ) nq+ + x i+ ξ i ) nq+} q x i+ x i f n) t) p dt p n! nq + ) q m { ξ i x i ) nq+ + x i+ ξ i ) nq+}) q m n! nq + ) q Finlly, observe tht x i+ x i f n) t) p dt p p p m f n) p { ξ i x i ) nq+ + x i+ ξ i ) nq+}) q f n) m n! nd Theorem 47 is proved h n i n! m h n i x i+ x i f n) t) dt The following corollry is consequence of Theorem 47 Corollry 48 Let f nd m be s defined bove The following estimtes pply i) The n th order left rectngle rule f t) dt m j ii) The n th order right rectngle rule m f t) dt j iii) The n th order trpezoidl rule f t) dt m n j +R T m, f), n h i ) j f j ) x i ) + R l m, f) j! n h i ) j f j ) x i+ ) + R r m, f) j! h ) j i { } f j ) x i ) ) j f j ) x i+ ) j!

119 A sofo where R l m, f) R r m, f) f n) m h n+ i, if f n) L, b n + )! f n) p n! nq + ) q m h nq+ i ) q, if f n) L p, b, p > nd p + q, f n) m h n i n!, if f n) L, b nd R T m, f) f n) m n n + )! f n) p n nq + ) q n! f n) m n! h n+ i, if f n) L, b, m h nq+ i ) q, if f n) L p, b, p > nd p + q, h n i, if f n) L, b Theorem 49 Consider the intervl x i α ) i ξ i α ) i x i+, i 0,, m, nd let f nd m be defined s bove Then we hve the equlity 80) m f t) dt j n ) j j! { ) j ξ i α ) i x i+ α ) i { ) j x i α i) i f j ) x i ) ξ i α ) i ) j } f j ) ξ i ) ) j f j ) x i+ )} + R ξ, α ) i, α ) i ), m, f

120 INTEGRAL INEQUALITIES FOR n TIMES DIFFERENTIABLE MAPPINGS nd the reminder stisfies the estimtion R ξ, α ) i, α ) i, m, f) f n) n + )! m { α ) i x i ) n+ + + α ) i ξ i ) n+ + ξ i α ) i ) n+ x i+ α ) i ) n+ }, if f n) L, b, f n) p n! nq + ) q m + { α ) i x i ) nq+ + α ) i ξ i ) nq+ + ξ i α ) i x i+ α ) if f n) L p, b, p > nd p + q, f n) m x i+ x i ) n, if f n) L, b n! i ) nq+ ) nq+ }) q, The proof follows directly from Theorem 9 on the intervls x i, x i+, i 0,, m ) The following Riemnn type formul lso holds Corollry 50 Let f nd m be defined s bove nd choose ξ i x i, x i+, i 0,, m ) Then we hve the equlity 8) f t) dt m j f j ) ξ i ) + R R ξ, m, f) n ) j {ξ j! i x i ) n+ ξ i x i+ ) n+} nd the reminder stisfies the estimtion R R ξ, m, f) f n) n + )! f n) p m n! nq + ) q ξ i x i ) n+ + x i+ ξ i ) n+), if f n) L, b, m { ξ i x i ) nq+ + x i+ ξ i ) nq+}) q, if f n) L p, b, p > nd p + q, f n) m h n i n!, if f n) L, b The proof follows from 80) where α ) i x i nd α ) i x i+

121 3 A sofo Remrk If in 8) we choose the midpoint ξ i x i+ + x i we obtin the generlised midpoint qudrture formul 8) f t) dt m j n ) j+ j! { ) j} f j ) xi + x i+ ) j hi ) + R M m, f) nd R M m, f) is bounded by R M m, f) f n) m n + )! n f n) p h n+ i, if f n) L, b, m ) q h nq+ n! nq + ) q n i, if f n) L p, b, p > nd p + q, m h n i, if f n) L, b f n) n! Corollry 5 Consider set of points 5xi + x i+ ξ i, x i + 5x i+ 6 6 i 0,, m ) nd let f nd m be defined s bove Then we hve the equlity b m n ) j hi ) j { 83) f t) dt ) j f j ) x i ) f j ) x i+)} j! 6 j { ξ i 5x ) j i + x i+ ξ 6 i x ) } j i + 5x i+ f j ) ξ 6 i ) + R s m, f) nd the reminder, R s m, f) stisfies the bound R s m, f)

122 INTEGRAL INEQUALITIES FOR n TIMES DIFFERENTIABLE MAPPINGS 4 f n) n + )! m f n) m { p n! nq + ) q xi + 5x i+ + 6 { ) n+ hi + ξ 6 i 5x ) n+ i + x i+ 6 ) } n+ xi + 5x i+ + ξ 6 i, if f n) L, b, ) nq+ hi + ξ 6 i 5x ) nq+ i + x i+ 6 ξ i ) nq+ }) q, if f n) L p, b, p > nd p + q, f n) m h n i n!, if f n) L, b Remrk If in 83) we choose the midpoint ξ i xi++xi we obtin generlised Simpson formul: f t) dt m j n ) j hi ) j { ) j f j ) x i ) f j ) x i+)} j! 6 hi 3 ) j { ) j} f j ) xi+ + x i ) + R s m, f) nd R s m, f) is bounded by R s m, f) f n) n + )! + f n) p n! f n) n! m n+) + n+ ) nq + ) n+ hi, if f n) L, b, 6 ) q m ) ) nq+ q hi, 6 if f n) L p, b, p > nd p + q, m h n i, if f n) L, b The following is consequence of Theorem 49 Corollry 5 Consider the intervl x i α ) i x i+ + x i α ) i x i+ i 0,, m ),

123 5 A sofo nd let f nd m be defined s bove The following equlity is obtined:- 84) f t) dt { xi+ + x i x i+ α ) i m j n ) j α ) i j! { ) j x i α i) i f j ) x i ) ) j xi+ + x i ) j f j ) x i+ )} + R B α ) ) } j α ) i f j ) xi+ + x i i, α ) i ), m, f, ) where the reminder stisfies the bound R B α ) i, α ) i, m, f) f n) n + )! f n) p m n! nq + ) q f n) m n! { α ) i + m h n i ) n+ xi+ + x i x i + α ) i x ) n+ i+ + x i + + { α ) i ) n+ α ) i x i+ α ) i ) nq+ xi+ + x i x i + ) nq+ + α ) i x i+ + x i ) n+ }, ) nq+ α ) i x i+ α ) i ) nq+ }) q, The following remrk pplies to Corollry 5 Remrk 3 If in 84) we choose we hve the formul: α ) i 3x i + x i+ 4 nd α ) i x i + 3x i+, 4 85) f t) dt m j n ) j j! ) j hi ) j f j ) x i ) f j ) x i+) 4 { ) j} f j ) xi+ + x i ) + R B m, f)

124 INTEGRAL INEQUALITIES FOR n TIMES DIFFERENTIABLE MAPPINGS 6 The reminder, R B m, f) stisfies the bound f n) m n + )! 4 ) n+ hi, 4 if f n) L, b f n) R B m, f) p 4 n! nq + ) q f n) m n! m ) ) nq+ q hi, if f n) L p, b, 4 p > nd p + q, h n i, if f n) L, b The following theorem incorportes the Newton-Cotes formul Theorem 53 Consider the intervl x i α ) i ξ ) i α ) i ξ ) i α 3) i x i+ i 0,, m ), nd let m nd f be defined s bove This considertion gives us the equlity m n ) j ) j ) j 86) f t) dt x i α ) i f j ) x i ) x i+ α 3) i j f j ) x i+ ) { ) j ξ ) i α ) i +R α ) i, α ) i, α 3) i, ξ ) j! { ξ ) i α ) i ξ ) i α 3) i i, ξ ) i, m, f ) j ) } j ξ ) i α ) i ) } j ) ) f j ) ξ ) i The reminder stisfies the bound R α ) i, α ) i, α 3) i, ξ ) i, ξ ) i, m, f) f n) n + )! + m f n) p n! nq + ) q + ) f j ) ξ ) i { ) n+ ) n+ ) n+ α ) i x i + ξ ) i α ) i + α ) i ξ ) i ξ ) i α ) i m ξ ) i α ) i ) n+ ) n+ + α 3) i ξ ) i + if f n) L, b, x i+ α 3) i ) n+ }, { ) nq+ ) nq+ ) nq+ α ) i x i + ξ ) i α ) i + α ) i ξ ) i ) nq+ ) nq+ + α 3) i ξ ) i + if f n) L p, b, p > nd p + q, f n) m h n i n!, if f n) L, b x i+ α 3) i ) nq+ }) q,

125 7 A sofo The following is consequence of Theorem 53 Corollry 54 Let f nd m be defined s bove nd mke the choices α ) i 7x i+x i+ 8, α ) i xi+xi+, α 3) i xi+7xi+ 8, ξ ) i xi+xi+ 3 nd ξ ) i xi+xi+ 3, then we hve the equlity: 87) m f t) dt j n ) j hi ) j { ) j f j ) x i ) f j ) x i+)} j! 8 ) j hi { 5 j 4) j} ) f j ) xi + x i+ 4 3 ) j hi { 4 j 5) j} ) f j ) xi + x i R N m, f), where the reminder stisfies the bound R N m, f) f n) n + )! 3 n+ + 4 n+ + 5 n+) m ) n+ hi, if f n) L, b, 4 f n) p n! 3 nq+ + 4 nq+ + 5 nq+ nq + ) q m ) ) nq+ q hi, 4 if f n) L p, b, p > nd p + q, f n) m h n i n!, if f n) L, b When n, we obtin from 87) the three-eighths rule of Newton-Cotes Remrk 4 For n from 87), we obtined perturbed three-eighths Newton-Cotes formul: b m hi ) ) hi f x i ) f ) x i+ ) f t) dt f x i ) + f x i+ )) ) ) )) 3hi xi + x i+ xi + x i+ + f + f ) ) ) 3hi f xi+x i+ 3 f xi+x i R N m, f),

126 INTEGRAL INEQUALITIES FOR n TIMES DIFFERENTIABLE MAPPINGS 8 where the reminder stisfies the bound R N m, f) f 9 m h 3 i, if f L, b 3 f q+ + 4 q+ + 5 q+ ) p q + f n! m q m ) ) q+ q hi, 4 if f L p, b, p > nd p + q, h i, if f L, b 8 Concluding Remrks This work hs subsumed, extended nd generlised mny previous Ostrowski type results Integrl inequlities for n times differentible mppings hve been obtined by the use of generlised Peno kernel Some prticulr integrl inequlities, including the trpezoid, midpoint, Simpson nd Newton-Cotes rules hve been obtined nd further developed into composite qudrture rules Moreover we hve brought together interior point rules giving explicit error bounds, using Peno type kernels nd results from the modern theory of inequlities Work on obtining bounds through the use of Peno kernels hs lso been, briefly treted, in the clssicl review books on numericl integrtion, by Stroud 3, Engels 3 nd Dvis nd Rbinowitz 7 However, in some other recent works, see for exmple Krommer nd Ueberhuber 6, constructive pproch is tken, vi Tylor or interpolting polynomils, to obtin qudrture results This pproch gives the order of pproximtion to the qudrture rule rther thn redily providing explicit error bounds Further reserch in this re will be undertken by considering the Chebychev nd Lupş inequlities Similrly, the following lternte Grüss type results my be used to exmine ll the interior point rules of this chpter Let σ h x)) h x) M g) where Then from 3) M h) b h t) dt T h, g) M hg) M h) M g)

127 Bibliogrphy P CERONE nd SS DRAGOMIR, Midpoint-type rules from n inequlities point of view, Hndbook of Anlytic-Computtionl Methods in Applied Mthemtics, Editor: G Anstssiou, CRC Press, New York 000), P CERONE nd SS DRAGOMIR, Trpezoidl-type rules from n inequlities point of view, Hndbook of Anlytic-Computtionl Methods in Applied Mthemtics, Editor: G Anstssiou, CRC Press, New York 000), P CERONE, SS DRAGOMIR nd J ROUMELIOTIS, An inequlity of Ostrowski type for mppings whose second derivtives re bounded nd pplictions, Est Asin J of Mth, 5 ) 999), -9 4 P CERONE, SS DRAGOMIR nd J ROUMELIOTIS, An inequlity of Ostrowski type for mppings whose second derivtives belong to L, b nd pplictions, Honm Mth J, ) 999), P CERONE, SS DRAGOMIR nd J ROUMELIOTIS, An Ostrowski type inequlity for mppings whose second derivtives belong to L p, b nd pplictions, submitted 6 P CERONE, S S DRAGOMIR nd J ROUMELIOTIS, Some Ostrowski type inequlities for n-time differentible mppings nd pplictions, Demonstrtio Mth, in press 7 PJ DAVIS nd P RABINOWITZ, Methods of Numericl Integrtion, nd Ed, Acdemic Press, New York, L DEDIĆ, M MATIĆ nd J PEČARIĆ, On some generlistions of Ostrowski inequlity for Lipschitz functions nd functions of bounded vrition, Mth Ineq & Appl, 3 ) 000), -4 9 SS DRAGOMIR, A generliztion of Ostrowski integrl inequlity for mppings whose derivtives belong to L, b nd pplictions in numericl integrtion, J of Computtionl Anlysis nd Applictions, in press) 0 SS DRAGOMIR, A generliztion of Ostrowski s integrl inequlity for mppings whose derivtives belong to L, b nd pplictions in numericl integrtion, SUT J of Mth, in press) SS DRAGOMIR, A generliztion of Ostrowski s integrl inequlity for mppings whose derivtives belong to L p, b, < p <, nd pplictions in numericl integrtion, J Mth Anl Appl, in press) SS DRAGOMIR, A Grüss type integrl inequlity for mppings of r-hölder s type nd pplictions for trpezoid formul, Tmkng Journl of Mthemtics, in press) 3 SS DRAGOMIR, New estimtion of the reminder in Tylor s formul using Grüss type inequlities nd pplictions, Mth Ineq & Appl, ) 999), SS DRAGOMIR, On the Ostrowski s integrl inequlity for mppings with bounded vrition nd pplictions, Mth Ineq & Appl, in press 5 SS DRAGOMIR, Ostrowski s Inequlity for Monotonous Mppings nd Applictions, J KSIAM, 3 ) 999), SS DRAGOMIR, The Ostrowski s integrl inequlity for Lipschitzin mppings nd pplictions, Comp nd Mth with Appl, ), S S DRAGOMIR, The Ostrowski integrl inequlity for mppings of bounded vrition, Bull Austrl Mth Soc, ), SS DRAGOMIR nd I FEDOTOV, An Inequlity of Grüss type for Riemnn-Stieltjes Integrl nd Applictions for Specil Mens, Tmkng J of Mth, 9 4) 998), SS DRAGOMIR nd S WANG, A new inequlity of Ostrowski s type in L p norm nd pplictions to some numericl qudrture rules, Indin J of Mth, 40 3) 998),

128 BIBLIOGRAPHY 0 0 SS DRAGOMIR nd S WANG, A new inequlity of Ostrowski type in L norm nd pplictions to some specil mens nd some numericl qudrture rules, Tmkng J of Mth, 8 997), SS DRAGOMIR nd S WANG, An inequlity of Ostrowski-Grüss type nd its pplictions to the estimtion of error bounds for some specil mens nd for some numericl qudrture rules, Computers Mth Applic, ), 5- SS DRAGOMIR nd S WANG, Applictions of Ostrowski s inequlity to the estimtion of error bounds for some specil mens nd to some numericl qudrture rules, Appl Mth Lett, 998), H ENGELS, Numericl Qudrture nd Cubture, Acdemic Press, New York, I FEDOTOV nd SS DRAGOMIR, Another pproch to qudrture methods, Indin J Pure Appl Mth, 308) 999), G GRÜSS, Über ds Mximum des bsoluten Betrges von b b ) f x) dx g x) dx, Mth Z, ), 5-6 b b f x) g x) dx 6 AR KROMMER nd CW UEBERHUBER, Numericl Integrtion on Advnced Computer Systems, Lecture Notes in Computer Science, 848, Springer-Verlg, Berlin, M MATIĆ, JE PEČARIĆ nd N UJEVIĆ, On new estimtion of the reminder in generlised Tylor s formul, Mth Ineq Appl, 999), DS MITRINOVIĆ, JE PEČARIĆ nd AM FINK, Clssicl nd New Inequlities in Anlysis, Kluwer Acdemic Publishers, A OSTROWSKI, Uber die Absolutbweichung einer differentienbren Funktionen von ihren Integrlmittelwert, Comment Mth Hel, 0 938), CEM PEARCE, J PEČARIĆ, N UJEVIĆ nd S VAROŠANEC, Generlistions of some inequlities of Ostrowski-Grüss type, Mth Ineq & Appl, 3 ) 000), AH STROUD, Numericl Qudrture nd Solution of Ordinry Differentil Equtions, Springer-Verlg, New York, 974

129 CHAPTER 3 Three Point Qudrture Rules by P CERONE nd SS DRAGOMIR Abstrct A unified tretment of three point qudrture rules is presented in which the clssicl rules of mid-point, trpezoidl nd Simpson type re recptured s prticulr cses Riemnn integrls re pproximted for the derivtive of the integrnd belonging to vriety of norms The Grüss inequlity nd number of vrints re lso presented which provide vriety of inequlities tht re suittble for numericl implementtion Mppings tht re of bounded totl vrition, Lipschitzin nd monotonic re lso investigted with reltion to Riemnn-Stieltjes integrls Explicit priori bounds re provided llowing the determintion of the prtition required to chieve prescribed error tolernce It is demonstrted tht with the bove clsses of functions, the verge of midpoint nd trpezoidl type rule produces the best bounds 3 Introduction Three point qudrture rules of Newton-Cotes type hve been exmined extensively in the literture In prticulr, the mid-point, trpezoidl nd Simpson rules hve been investigted more recently 33-0 with the view of obtining bounds on the qudrture rule in terms of vriety of norms involving, t most, the first derivtive The bounds tht hve been obtined more recently lso depend on the Peno kernel used in obtining the qudrture rule The generl pproch used in the pst involves the ssumption of bounded derivtives of degree higher thn one The prtitioning is hlved until the desired ccurcy is obtined see for exmple Atkinson ) The work in ppers 33-0 ims t obtining priori estimtes of the prtition required in order to ttin prticulr bound on the error The current work employs the modern theory of inequlities to obtin bounds for three-point qudrture rules consisting of n interior point nd boundry points The mid-point, trpezoidl nd Simpson rules re recptured s prticulr instnces of the current development Riemnn integrls re pproximted for the derivtive of the integrnd belonging to vriety of norms An inequlity due to Grüss together with number of extensions nd vrints is used to obtin perturbed

130 3 THREE POINT QUADRATURE RULES three-point rules which produce tight bounds suitble for numericl qudrture The pproximtion of Riemnn-Stieltjes integrls is lso investigted for which the mppings belong to vriety of clsses including: totl bounded vrition, Lipschitzin nd monotonic The chpter is divided in two sections The first contins estimtes of the error in terms of t most the first derivtive while the second dels with the reminder when the function is n - differentible The first section is rrnged in the following mnner In Subsection 3, n identity is derived tht involves three-point rule whose bound my be obtined in terms of the first derivtive, f L, b Appliction of the results in numericl integrtion re presented in Subsection 3 An Ostrowski- Grüss inequlity is developed in Subsection 33, s is premture Grüss which produces perturbed three-point rules A further Ostrowski-Grüss inequlity is developed in Subsection 34 which produces bounds tht re even shrper thn those obtined from the premture Grüss results Results nd numericl implementtion of inequlities in which the first derivtive f L, b re developed in Subsection 35, while perturbed three-point rules re obtined in Subsection 36 through the nlysis of some new Grüss-type results Three-point Lobtto rules re obtined in Subsection 37 when f L p, b, while perturbed rules through the development of Grüss-type rules re investigted in Subsection 38 Subsection 39 is reserved for functions tht re not necessrily differentible nd so inequlities involving Riemnn-Stieltjes integrls tht re suitble for numericl implementtion re investigted in which the functions re ssumed to be either of totl bounded vrition, Lipschitzin or monotonic The work repetedly demonstrtes tht Newton-Cotes rule tht is equivlent to the verge of mid-point nd trpezoidl rule consistently gives tighter bounds thn Simpson-type rule Some concluding remrks to the section nd discussion re given in Subsection 30 The second section is structured s follows A vriety of identities re obtined in Subsection 33 for f n ) bsolutely continuous for generlistion of the kernel 398) Specific forms re highlighted nd generlised Tylor-like expnsion is obtined Inequlities re developed in Subsection 333 nd perturbed results through Grüss inequlities nd premture vrints re discussed in Subsection 334 Subsection 335 demonstrtes the pplicbility of the inequlities to numericl integrtion Concluding remrks to the section re given in Subsection 337

131 3 P Cerone nd SS Drgomir 3 Bounds Involving t most First Derivtive 3 Inequlities Involving the First Derivtive We strt with the following results in terms of sup-norms Theorem 3 Let f :, b R be differentible mpping on, b) whose derivtive is bounded on, b) nd denote f sup t,b) f t) < Further, let α :, b R nd β :, b, b, α x) x, β x) x Then, for ll x, b we hve the inequlity 3) f t) dt β x) α x)) f x) + b β x)) f b) + α x) ) f ) { f + b α x) + x ) + x + b ) ) + β x) b + x ) } Proof Let 3) K x, t) { t α x), t, x t β x), t x, b, nd consider Now, from 3), K x, t) f t) dt K x, t) f t) dt x t α x)) f t) dt + x nd integrting by prts produces the identity 33) K x, t) f t) dt t β x)) f t) dt, β x) α x)) f x) + b β x)) f b) + α x) ) f ) f t) dt Thus, 34) f t) dt β x) α x)) f x) + b β x)) f b) + α x) ) f ) f K x, t) dt

132 Let Q x) K x, t) dt nd so αx) x Q x) t α x)) dt + βx) x 3 THREE POINT QUADRATURE RULES 4 αx) t α x)) dt t β x)) dt + t β x)) dt βx) { α x)) + x α x)) + x β x)) + b β x)) } If we use the identity X + Y X + Y 35) we my write Q x) s x 36) Q x) ) + α x) + x ) ) X Y + ) ) b x + + β x) b + x ) x ) + b x) in 36) nd substitution The reutilizing of identity 35) on into 34) will produce the result 3) nd thus the theorem is proved Corollry 3 Let f stisfy the conditions of Theorem 3 Then α x) +x nd β x) b+x give the best bound for ny x, b nd so 37) f f t) dt b b f x) + x b ) ) + x + b ) f ) + b x b ) f b) Proof The proof is trivil since 3) is sum of squres nd the minimum occurs when ech of the terms re zero Remrk 3 Result 37) is similr to tht obtined by Milnović nd Pečrić 45, p 470, lthough their bound relies on the second derivtive being bounded This is not lwys possible so tht the weker ssumption of the first derivtive being bounded s in 34) my prove to be useful Remrk 3 An even more ccurte qudrture formul is obtined when x +b, giving from 37) : 38) f f t) dt b ) b f + b ) + f ) + f b) This is equivlent to pproximting n integrl s the verge of mid-point nd trpezoidl rule The bound in 37) however, only requires the first derivtive of the function f to be bounded

133 5 P Cerone nd SS Drgomir Motivted by the results of Theorem 3 nd Corollry 3, we consider kernel of the form 3) where α x) nd β x) re convex combintions of the end points Theorem 33 Let f stisfy the conditions s stted in Theorem 3 Then the following inequlity holds for ny γ 0, nd x, b : f t) dt b ) { γ) f x) ) ) x b x + γ f ) + f b)} b b f 4 + γ ) b ) + x + b ) 39) 30) b ) f Proof Let 3) α x) γx + γ) nd β x) γx + γ) b Thus, from Theorem 3 nd its proof utilizing 33) where 3) Q x) K x, t) dt we hve from 35), on substituting for α x) nd β x) from 3), tht ) x Q x) + γ ) ) b x x ) + + γ ) b x) Thus, 33) Q x) γ ) x ) + b x) γ ) b ) + x + b ), upon using the identity 35) Now, utilizing 3) nd 33) in 34) will give the first prt of the theorem, nmely, eqution 38) Inequlity 30) cn esily be scertined since 39) ttins its mximum t γ 0 or, nd t x or b Remrk 33 Corollry 3 my be recovered of γ is set t its optiml vlue of in Theorem 33 Remrk 34 γ 0 in 39) reproduces Ostrowski s inequlity 45, p 468 whose bound is shrpest when x +b, giving the midpoint rule Remrk 35 γ produces the generlized trpezoidl rule for which gin the best bound occurs when x +b giving the stndrd trpezoidl type rule

134 3 THREE POINT QUADRATURE RULES 6 Corollry 34 Let the conditions on f be s in Theorem 3 Then, the following inequlity holds 34) f { f t) dt b ) γ) f b ) 4 + γ ) + b ) + γ f ) + f b) } Proof Plcing the optiml vlue of x +b in 38) produces the result 34) Remrk 36 Result 34) gives liner combintion between mid-point nd trpezoidl rule The optiml result is obtined by tking γ in 34), giving the optiml bound when only the ssumption of bounded first derivtive is used This gives the result from 34) f t) dt b ) + b f ) + f b) 35) f + b ) 8 f, which is equivlent to 38) It should be noted tht tking γ 3 in 34) gives Simpson-type rule tht is worse thn 35), remembering tht here we re only using the ssumption of bounded first derivtive rther thn the more restrictive though more ccurte) result of bounded fourth derivtive 3 Appliction in Numericl Integrtion The following qudrture result holds Theorem 35 Let f :, b R be differentible mpping on, b) with f sup t,b) f t) < Then for ny prtition I n : x 0 < x < < x n < x n b of, b nd ny intermedite point vector ξ ξ 0, ξ,, ξ n ) such tht ξ i x i, x i+ for i 0,,, n, we hve: 36) where f x) dx A c f, I n, ξ) + R c f, I n, ξ), A c f, I n, ξ) n γ) h i f ξ i ) + γ n γ) h i f ξ i ) + γ n n n ξ i x i ) f x i ) + x i+ ξ i ) f x i+ ) ξ i f x i+ ) f x i )) + bf b) f )

135 7 P Cerone nd SS Drgomir nd the reminder R c f, I n, ξ) f 4 + γ ) n hi ) + ξ i x ) i + x i+ f n where ν h) mx,,n h i h i f nν h), Proof Applying inequlity 38) on the intervl x i, x i+ for i 0,,,, n we hve xi+ f x) dx x i { γ) f ξ i ) h i + γ ξ i x i ) f x i ) + x i+ ξ i ) f x i+ )} f 4 + γ ) hi ) + ξ i x ) i + x i+ f h i, since the corsest bound is obtined t γ 0 or nd ξ i x li, x i+ Summing over i for i 0 to n we my deduce 36) nd its subsequent elucidtion Corollry 36 Let the ssumptions of Theorem 35 hold Then we hve where n A c f, I n ) γ) h i f f x) dx A c f, I n ) + R c f, I n ) xi + x i+ R c f, I n ) f ) + γ n h i f x i ) + f x i+ )), which is liner combintion of the mid-point nd trpezoidl rules nd the reminder R f, I n ) stisfies the reltion where ν h) mx,,n h i f γ ) n h i γ ) nν h), Proof Similr to Theorem 35 with ξ i xi+xi+

136 3 THREE POINT QUADRATURE RULES 8 33 A Generlized Ostrowski-Grüss Inequlity Using Cuchy-Schwrtz The following result is known s the Grüss inequlity which ws proved by Grüss in 935 see for exmple 44, p 96) Theorem 37 Let f, g be two integrble functions defined on, b, stisfying the conditions c f t) C nd d g t) D for ll t, b Then 37) T f, g) C c) D d), 4 where 38) T f, g) b nd the constnt 4 is the best possible f t) g t) dt b f t) dt b g t) dt, The proof of this theorem is n extension of tht for Theorem 34 nd discussion will be delyed until then See lso Remrk 34 Theorem 38 Let f : I R R be differentible mpping in I nd let, b I with < b Further, let f L, b nd d f x) D, x, b We hve, then, the following inequlity { ) x f t) dt b ) γ) f x) + γ f ) b ) } b x + f b) + b ) γ) x + b ) S b { D d) b b ) + x + b + γ ) 39) b ) 4 + x + b γ ) } b ) where S fb) f) b nd γ 0, Proof From the identity 33) with α x) nd β x) s defined in 3) we hve { ) ) } x b x 30) f t) dt b ) γ) f x) + γ f ) + f b) b b where K x, t) f t) dt 3) K x, t) { t γx + γ), t, x t γx + γ) b, t x, b Now it is cler tht for ll x, b nd t, b we hve tht φ x) K x, t) Φ x)

137 9 P Cerone nd SS Drgomir where nd φ x) mx {γ x ), γ) b x)} Φ x) mx { γ) x ), γ b x)} Using the result tht mx {X, Y } X+Y + Y X we hve tht Φ x) γb γ) + γ) x + γb + γ) x nd Thus, 3) Now, φ x) γ ) x + γ) b γ + γ + γ) b x Φ x) φ x) b + { x + b + γ ) b ) + x + b γ ) } b ) 33) x K x, t) dt t γx + γ) dt + γ)x ) γx ) γb x) udu + x γ)b x) t γx + γ) b dt vdv γ) γ x ) b x) b ) γ) x + b ) Applying Theorem 37 of Grüss to the mppings K x, ) nd f t), nd using S b b f t) dt, we obtin from 3) nd 33), b K x, t) f t) dt γ) x + b ) S b { D d) b + x + b + γ ) b ) 4 + x + b γ ) } b ) Then, using the identity 30) we obtin the result 39) s stted in the theorem Hence, the theorem is completely proved Remrk 37 It should be noted tht the shifted qudrture rule tht is obtined through the Grüss inequlity still involves function evlutions t the end points

138 3 THREE POINT QUADRATURE RULES 30 nd n interior point x Thus, simple grouping of terms would produce the left hnd side of 39) in n lternte form + { f t) dt γ x ) γ) b ) f x) + x + b ) } f b) γ b x) + x + b ) f ) Therefore, it is rgued, the bove qudrture rule is no more difficult to implement thn the rule s given in Theorem 33 for exmple Corollry 39 Let the conditions be s in Theorem 38 Then the following inequlity holds for ny x, b, b f t) dt 34) b ) f x) + x ) f ) + b x) f b) D d) b ) b + 4 x + b Proof Setting γ in 39) redily produces the result 34) Corollry 30 Let the conditions be s in Theorem 38 Then the following inequlity holds for ny γ 0,, 35) { f t) dt b ) γ) f D d) b ) 4 + b + γ ) ) + γ f ) + f b) } Proof Choosing x to be t the mid-point of, b in 39) gives the result 35) Remrk 38 Plcing γ 0 in 39) produces n djusted Ostrowski type rule, nmely: f t) dt f x) x + b ) S D d) 8 b ) + x b + x This bound is shrpest t x +b, thus producing the mid-point type rule 36) + b f t) dt b ) f ) D d) b ) 4

139 3 P Cerone nd SS Drgomir Remrk 39 Plcing γ in 39) gives n djusted generlized trpezoidl rule, nmely: b ) ) x b x f t) dt f ) + f b) x + b ) S b b b D d) 8 b + x b + x This bound is shrpest t x +b giving the trpezoidl type rule 37) f t) dt b D d) b ) f ) + f b) 4 Remrk 30 The shrpest bound on 34) nd 35) re t x +b nd γ respectively The sme result cn be obtined directly from 39), giving s the best qudrture rule of this type 38) f t) dt b ) f + b ) + b ) D d) 8 which is n verged mid-point nd trpezoidl rule f ) + f b) Agin, s noted in Subsection 3 when it ws ssumed tht f L, b), γ 3 in 35)) produces Simpson type rule which is worse thn the optiml rule given by 38) Here, we re only ssuming tht d < f x) < D rther thn the more restrictive, though more ccurte, ssumptions in the development of trditionl Simpson s rule of bounded fourth derivtive The following two results by Ostrowski will be needed for the proof of the theorem tht follows An improvement by Lups is lso presented These will be presented s theorems which re generliztions of the Grüss inequlity The nottion of Pečrić, Proschn nd Tong 47 will be used Theorem 3 Let f be bounded mesurble function on I, b) such tht c f t) c for t I nd ssume g t) exists nd is bounded on I Then, nd 8 T f, g) b 8 is the best constnt possible c c ) sup g t) t I Theorem 3 Let g be loclly bsolutely continuous on I with g L I), nd let f be bounded nd mesurble on I, b) such tht c f t) c for t I Then 39) T f, g) b 4 c c ) g, nd, the improved result, 330) T f, g) b π c c ) g,

140 3 THREE POINT QUADRATURE RULES 3 where g b g t) dt ) Theorem 33 Let f : I R R be differentible mpping on I nd let, b I with < b Further, let f L, b nd d f x) D, x, b Then, the following inequlity holds 33) f t) dt b ) + b ) γ) x + b ) S D d b ) 8 where S fb) f) b nd γ 0, { γ) f x) + γ ) x f ) + b ) } b x f b) b Proof Let K x, t) be s given by 3) nd consider the intervl, x Let d f t) D for t, x Then, from Theorem 37, T,x f, K) x D d ), 8 since sup K x, t), s K t,x Let d f t) D for t x, b Then, in similr fshion Tx,b f, K) b x D d ) 8 Now, using the tringle inequlity redily produces T,b f, K) x ) D d ) + 8 b 8 D d) b x) 8 D d ) Thus, from 33) nd 3), T,b f, K) b K x, t) f t) dt γ) x + b ) S b b D d) 8 Using identity 30) redily produces 33), nd the theorem is proved Remrk 3 On ech of the intervls, x nd x, b sup k t) k, t I where k t) K x, t) Thus, using Theorem 3 is superior to using either of the two results of Theorem 3

141 33 P Cerone nd SS Drgomir Remrk 3 The bound obtined by 33) is uniform The bound given by 39) ttins its shrpest bound when x +b nd γ D d), producing 8 b ) Thus, the current bound is better for b > nd for ll x Remrk 33 If Theorem 3 is used nd T K, f ) is considered, then result similr to 39) would be obtined with D d 4 being replced by f 8 This will not be investigted further since the second derivtive is involved, thus plcing it outside the scope of the work of this section The following result shll be termed s premture Grüss inequlity in tht the proof of the Grüss inequlity is not tken to its finl conclusion but is stopped premturely Theorem 34 Let f, g be integrble functions defined on, b, nd let d g t) D Then 33) T f, g) D d T f, f), where 333) T f, f) M f M f) with 334) M f) b nd is the best possible constnt f t) dt Proof The proof follows tht of the Grüss inequlity s given in 44, p 96 The identity 335) T f, g) b ) f t) f τ)) g t) g τ)) dtdτ my esily be shown to be vlid Now, pplying the Cuchy-Schwrtz-Bunikowsky integrl inequlity for double integrls, we hve, on denoting the right hnd side of 335) by T f, g), Therefore, from 335) T f, g) T f, f) T g, g) 336) T f, g) T f, f) T g, g) Now, 337) T g, g) D M g)) M g) d) b D M g)) M g) d) D g t)) g t) d) dt since d g t) D In ddition, using the elementry inequlity for ny rel numbers p nd q, ) p + q pq,

142 3 THREE POINT QUADRATURE RULES 34 we hve, from 337), ) D d 338) T g, g) Combining 338) nd 336), the results ) re obtined nd the theorem is proved To prove the shrpness of 33) simply tke f t) g t) sgn ) t +b Remrk 34 To prove 37), the bound ) C c for T f, f) would be obtined in similr fshion to tht of T g, g), nd hence the term premture Theorem 35 Let the conditions be s in Theorem 38 The following shrper inequlity holds: 339) { f t) dt b ) γ) f x) + γ ) } b x + f b) b { b ) D d) b ) γ + 3 x + b ) 4 γ ) } x b b ) γ) x + b ) ) f ) ) S where S fb) f) b, the secnt slope Proof The proof of the current theorem follows long similr lines to tht of Theorem 38 with the exception tht premture Grüss theorem Theorem 3) is used From the identity 30) the function K x, t) is known nd it is s given by 3) Thus, pplying the premture Grüss theorem Theorem 3) to the mppings K x, ) nd f ) we obtin 340) b D d K x, t) f t) dt b b K x, t) dt b K x, t) dt f t) dt b ) b K x, t) dt

143 35 P Cerone nd SS Drgomir Now, from 3), b b b 3 b ) K x, t) dt { x t γx + γ) ) dt + { γ)x ) γx ) The well-known identity 34) X 3 + Y 3 X + Y ) my be utilized to give 34) 3 γb x) u du + v dv γ)b x) γ 3 + γ) 3 x ) 3 + b x) 3 b Thus, using the fct tht K x, t) dt γ b X + Y x t γx + γ) b) dt } ) ) X Y + 3 ) b ) + 3 x + b ) f t) dt f b) f ) b S, the secnt slope together with 33) nd 34) gives, from 340) : b K x, t) f t) dt γ) x + b ) 343) S b { D d) γ ) b ) + 3 x + b ) 4 γ ) x + b { b D d) 3 ) } ) γ ) + 3 x + b ) 4 γ ) } The term in the brces is, of course, positive since b >, γ 0,, x, b nd 4 γ γ γ) ) Utilizing the identity 30) in 343) produces the result 339) Thus, the theorem is proved }

144 3 THREE POINT QUADRATURE RULES 36 Corollry 36 Let the conditions be s in Theorem 38 Then the following inequlity holds for ll x, b, b f t) dt 344) b ) f x) + x ) f ) + b x) f b) b ) D d) 4 b ) + 3 x + b ) 3 Proof The result 344) is redily obtined from 339) by substituting γ Corollry 37 Let the conditions be s in Theorem 38 Then the following inequlity holds for ll γ 0, { ) + b f t) dt b ) γ) f + γ } 345) f ) + f b) D d) 4 b ) γ ) Proof Tking x +b in 339) together with minor rerrngement gives 345) Remrk 35 Result 339) is shrper thn result 39) since the premture Grüss theorem is shrper thn the Grüss theorem utilized to obtin 39) Remrk 36 Substituting γ 0 into 339) gives n djusted Ostrowski type rule, nmely f t) dt b ) f x) x + b ) S D d) 4 b ) 3 This is uniform bound which does not depend on the vlue of x Thus, midpoint rule would hve the sme bound s evluting the function t ny x, b together with n djustment fctor Evlution of the bove result t x or x b produces the stndrd trpezoidl rule Remrk 37 Tking γ in 339) gives x f t) dt b ) b D d) 4 b ) 3 ) f ) + ) b x f b) + x + b ) S b Tht is, using the fct tht S fb) f) b, the trpezoidl rule, f t) dt b D d) f ) + f b) 4 b ), 3 is recovered

145 37 P Cerone nd SS Drgomir Remrk 38 The shrpest bounds for 344) nd 345) re t x +b nd γ respectively This result cn be obtined directly from 339) by tking x nd γ t the mid-point, giving the best qudrture rule of this type s + b 346) f t) dt b ) f ) + f ) + f b) D d) 8 b ) 3 If γ 3 is tken in 345), then Simpson type rule is obtined, giving f t) dt b ) + b f ) + f b) D d) f b ) 3 3 ) This bound is worse thn the optiml rule 346) by reltive mount of 3 which is pproximtely 55% Computtionlly, the qudrture rule 346) is just s esy to pply s Simpson s rule since the only difference is the weights Remrk 39 The bound in 346) is 3 times better thn tht in 38) Tht is, ) the bound in 38) is worse thn tht in 346) by reltive mount of 3 The optiml qudrture rule of this subsection will now be pplied from 346) nd it will be denoted by A o Theorem 38 Let f : I R R be differentible mpping in I the interior of I) nd let, b I with b > Let f L, b nd d f x) D, x, b Further, let I n be ny prtition of, b such tht I n : x 0 < x < < x n < x n b Then we hve where nd f x) dx A o f, I n ) + R o f, I n ) A o f, I n ) n ) xi + x i+ h i f + n h i f x i ) + f x i+ ), 4 with ν h) mx,,n h i R o f, I n ) n D d) 8 3 h i D d) 8 3 nν h) Proof Applying inequlity 346) on the intervl x i, x i+ for i 0,,, n we hve xi+ f x) dx h ) i xi + x i+ D d) f + f x i ) + f x i+ ) h i x i where h i x i+ x i Summing over i for i 0 to n gives A o f, I n ) nd R o f, I n )

146 3 THREE POINT QUADRATURE RULES 38 Corollry 39 Let the conditions of Theorem 38 hold In ddition, let I n be the equidistnt prtition of, b, I n : x i + ) b n i, i 0,,, n then D d) f x) dx A o f, I n ) 8 b ) 3 n Proof From Theorem 38 with h i b n n D d) R o f, I n ) 8 b 3 n nd hence the result is proved for ll i such tht ) D d) 8 3 b ) n Remrk 30 If we wish to qpproximte the integrl f x) dx using the qudrture rule A o f, I n ) of Corollry 39 with n ccurcy of ε > 0, then we need n ε N points for the equispced prtition I n where D d) b ) n ε ε where x denotes the integer prt of x It should further be noted tht the ppliction of Corollry 39, in prctice, is costly s it stnds The following corollry is more pproprite s it is more efficient Corollry 30 Let the conditions of Theorem 38 hold nd let I m be the equidistnt prtition of, b, I m : x i +ih, i 0,,, m with h b m Then f x) dx h 4 f x 0) + f x m ) h m f x i ) D d 6 b ) 3 m since Now, m Thus Proof From Theorem 38 A o f, I m ) h f xi ) + f x i+) ) m x i+) + x i f x i+ ) + h 4 i m + h i + ) x i+ f xi ) + f x i+) ) m m f x 0 ) + f x m ) + f x i ) + f ) x i+) i m f x 0 ) + f x m ) + f x i ) A o f, I m ) h 4 f x 0) + f x m ) + h i m i f x i ),

147 39 P Cerone nd SS Drgomir where h b m Further, from Theorem 38 with h i b m D d) R o f, I m ) 8 3 The corollry is thus proved m for i 0,,, m, ) D d b m 6 3 b ) m 34 A Generlized Ostrowski-Grüss Inequlity Vi New Identity Trditionlly, the Grüss inequlity ws effectively obtined by seeking bound on T f, g) vi double integrl identity nd the Cuchy-Schwrtz-Bunikowsky integrl inequlity to reduce the problem down to obtining bounds for T f, f) Recently, Drgomir nd McAndrew 30 hve obtined bounds on T f, g) s defined in 38), where f nd g re integrble, by using the identity 347) T f, g) b Hence 348) T f, g) b f t) M f) g t) M g) dt f t) M f)) g t) M g)) dt In prticulr, they pply the inequlity when one of the functions is known nd so effectively lthough not explicitly stted) use 349) T f, g) sup g t) M g) t,b b where f is known f t) M f) dt, Theorem 3 Let f : I R R be differentible mpping in I nd let, b I with < b Furthermore, let f L, b nd d f x) D, x, b Then the following inequlity holds: 350) f t) dt b ) + b ) γ) x + b ) S D d + S d + D I γ, x) D d) I γ, x), where 35) I γ, x) { γ) f x) + γ 35) K x, t) ) x f ) + b K x, t) γ) x + b ) dt, { t γx + γ) ), t x t γx + γ) b), t > x, γ 0,, ) } b x f b) b

148 3 THREE POINT QUADRATURE RULES 40 nd S f b) f ) b Proof Applying 349) on the mppings K x, ) nd f ) gives 353) b ) T K, f ) sup f b t) S K x, t) K x, u) du t,b b dt Now, nd, from 33), nd so mx {D S, S d} D d b + S d + D K x, u) du γ) x + b ), K x, t) b K x, u) du dt I γ, x) Hence, D d 354) T K, f ) + S d + D I γ, x) f t) dt, Furthermore, using identity 30), 354) nd the fct tht S b 350) results nd the first prt of the theorem is proved Tking S d or D provides the upper bound given by the second inequlity We now wish to determine closed form expression for I γ, x) s given by 35) where K x, t) is from 35) Now, I γ, x) my be written s 355) I γ, x) where x t φ x) dt + x t ψ x) dt 356) φ x) γ) x + γb b b, ψ x) γ) x + γ + In 355), let b ) u t φ x) nd b ) v t ψ x), such tht { x φx) ψx) } 357) I γ, x) b ) b b u du + v dv φx) b x ψx) b To simplify the problem it is worthwhile to prmeterize the prtitioning of the intervl, b To tht end let x δb + δ), δ 0, so tht 358) δ x b, δ b x b nd x + b b ) δ )

149 4 P Cerone nd SS Drgomir Now, from 356) 359) φ x) b +b x b ) γ ) b x b 360) 36) x φ x) b δ γ δ) γ) γ) δ, x ψ x) b ) b x γ b δ) γ γ δ γ γ ), ) x b γδ γ δ γ nd 36) b ψ x) b +b x ) + γ b ) x b δ + γδ γ) γ) δ Thus, 357) becomes, for x δb + δ), on using 359) 36), 363) I γ, x) J γ, δ) b ) J γ, δ) + J γ, δ), where 364) J γ, δ) nd 365) J γ, δ) It should be noted tht nd γδ γ ) u du γ) δ γ) γ) δ γ) γδ γ J γ, δ) J γ, δ) v dv 366) J γ, δ) J γ, δ),

150 3 THREE POINT QUADRATURE RULES 4 so tht only one of 364) or 365) need be evluted explicitly nd the other my be obtined in terms of it We shll consider J γ, δ) in some detil There re three possibilities to investigte The limits in 365) re either both negtive, one negtive nd one positive, or re both positive We note tht the top limit is lwys greter thn the bottom since γ) δ > γδ Thus, over the three different regions we hve: J γ, δ) γ) δ γ) γδ γ 0 γδ vdv, γ) < δ < γ γ) δ γ vdv + γ) 0 vdv, δ < γ, δ < γ) γ) δ γ) γδ γ γδ vdv, γ < δ < γ) ) γ) δ), γ) < δ < γ γδ ) + γ) δ), δ < γ, δ < γ) γ) δ) ) γδ, γ < δ < γ) Now, using the result X Y my be simplified to give 367) J γ, δ) X Y ) X + Y ) nd 35), the bove expressions δ) γ ) δ, γ) < δ < γ δ ) ) + γ δ, δ < γ, δ < γ) δ) γ ) δ, γ < δ < γ) Further, using 366), n expression for J γ, δ) my be redily obtined from 367) to give 368) J γ, δ) δ γ) δ), γ) < δ < γ δ ) + γ) δ), δ > γ, δ > γ) δ γ) δ), γ < δ < γ) For n explicit evlution of I γ, x), 363) needs to be determined This involves the ddition of J γ, δ) nd J γ, δ) This my best be ccomplished by reference to digrm Figure 3 shows five regions on the γδ plne defined by the curves δ γ), δ γ, δ γ, δ γ), where γ 0, γ, δ 0, δ

151 43 P Cerone nd SS Drgomir define the outside boundry The regions re defined s follows A : δ > γ), δ < γ, δ > γ, δ > γ) ; 369) B : δ > γ, δ < γ), δ > γ, δ > γ) ; C : δ < γ), δ < γ, δ < γ, δ > γ) ; D : δ < γ, δ < γ), δ < γ), δ > γ ; nd E : δ < γ), δ < γ, δ < γ, δ > γ) It is importnt to note tht the first two inequlities in ech of the regions define the contributions from J γ, δ) nd the second two, tht of J γ, δ) Thus using 367) 369), 363) is given by ) γ δ) γ ) δ) δ + δ ) on A 370) J γ, δ) b ) ) γ δ) γ ) δ) + δ + δ ) on B γ ) δ γ ) δ + δ) + δ ) on C γ ) δ γ 4 + ) γ 4 + ) δ ) δ δ) + δ ) on D on E Remrk 3 We my now proceed in one of two wys One pproch is to trnsform to n expression involving x, thus giving I γ, x), nd so 350) my be used The second pproch is to work in terms of δ so tht Theorem 3 would be converted to n expression involving δ We will tke modifiction of the first pproch Once prticulr vlue γ is determined which dicttes the type of rule δ is trnsformed in terms of x using the reltion J γ, δ) I γ, x), where δ x b Remrk 3 Tking different vlues of γ will produce bounds for vrious inequlities For γ 0, then from Figure 3 it my be seen tht we re on the left boundry of region A nd D nd obtin, from 370), uniform bound independent of δ, J 0, δ) b ) 4 Thus, from 350), perturbed Ostrowski inequlity is obtined on noting from 363) tht I γ, x) J γ, δ), 37) f t) dt b ) f x) x + b b ) 4 D d + S d + D ) S

152 3 THREE POINT QUADRATURE RULES A B E 0 D C Figure 3 Digrm showing regions of vlidity for Jγ,δ) b ) given by 369) nd 370), s well s its contours s For γ, it my be noticed from Figure 3 tht we re now on the right boundry of B nd D so tht from 370), uniform bound independent of δ is obtined viz, J, δ) b ) 4 Thus, from 350), perturbed generlized trpezoidl inequlity is obtined, nmely 37) b ) 4 f t) dt b ) D d + ) ) x b x f ) + f b) x + b ) S b b S d + D Tking x +b reproduces the result of Drgomir nd McAndrew 30 Agin, it my be noticed tht the bove result is uniform bound for ny x, b

153 45 P Cerone nd SS Drgomir Corollry 3 Let the conditions of f be s in Theorem3 Then the following inequlity holds for ny x, b : b f t) dt 373) b ) f x) + x ) f ) + b x) f b) b ) + x + b ) D d + 4 S d + D, where S fb) f) b Proof Letting γ in 350) redily produces the result 373) from 370), on noting tht I, x) J, δ) b ) δ where b ) δ ) x +b Corollry 33 Let the conditions of f be s in Theorem 3 Then the following inequlity holds for ny γ 0, : 374) b ) 4 { f t) dt b ) γ) f 4 + γ ) D d + b + S d + D ) + γ f ) + f b) } Proof Letting x +b in 350) produces the result 374) from 370) on noting I ) ) γ, +b J γ, b ) ) γ in region E Remrk 33 Tking x +b in 373) or γ in 374) is equivlent to tking both these vlues in 350) This produces the shrpest bound in this clss, giving f t) dt b { ) + b f + } 375) f ) + f b) b ) D d + 6 S d + D For the bound, it is equivlent to tking the point, ) in region E from 370) nd Figure 3, thus giving J, ) 6 For Simpson type rule, tking the point 3, ) in region E from 370), nd Figure 3 gives J 3, ) which is corser bound thn J, ) t which the minimum occurs the centre point in Figure 3) Remrk 34 It should be noted tht the best bound possible with the premture Grüss is given by 346) This my be compred with the current bound 375) Now, 375) is computtionlly more expensive, but even the worst bound, b ) 6 D d) in 375) is better thn tht of 346)

154 3 THREE POINT QUADRATURE RULES 46 Remrk 35 A generlized Simpson type rule my be obtined by tking γ 3 for unprescribed x Thus, from 370), δ 6 +5δ 6 + ) δ 3 4 < δ < 376) J 3, δ) b ) 5 8 nd so from 350) : b ) 377) f t) dt 3 + b x + b ) S 3 D d + S d + D where with Tht is, from 376), ) I 3, x 4 + ) δ 4 < δ < 3 4 δ 6 6 5δ 6 + ) δ 0 < δ < 4 f x) + ) I 3, x, ) x f ) + b ) ) I 3, x b ) J 3, δ δ x b, ) b x f b) b b x) b )+5x ) ) x, 3 4 < x b 5 b ) ) 8 + x +b, 4 < x b < 3 4 x 6 6b ) 5x ) 6 + ) b x, 0 < x b < 4 Remrk 36 It my hve been noticed from Figure 3 or, for tht mtter, directly from 370) Replcing δ by δ in A nd B would give the regions D nd C respectively Also, replcing γ by γ in B nd C would give the regions A nd D respectively Thus, it would hve been possible to investigte the region γ nd δ since we my redily trnsform ny point γ, δ ) in the γδ plne to one in this region Thus, only the regions B nd E would need to be nlyzed where for γ, δ, B : δ > γ, nd E : δ < γ This pproch ws not followed since we were more interested in evlution long lines perpendiculr to the xes Remrk 37 For prcticl implementtion of the bove in numericl integrtion it would be expensive to clculte the bounds s given However, insted of D d + S d+d being used, the corser bound of D d my be more suitble

155 47 P Cerone nd SS Drgomir The optiml qudrture rule of this subsection will now be pplied from 375) nd it will be denoted by A 0 Theorem 34 Let f : I R R be differentible mpping on I the interior of I) nd let, b I with b > Let f L, b nd d f x) D, x, b In ddition, let I n be prtition of, b such tht I n : x 0 < x < < x n < x n b Then we hve where nd where nd f x) dx A 0 f, I n ) + R 0 f, I n ), A 0 f, I n ) n ) xi + x i+ h i f + n h i f x i ) + f x i+ ) 4 R 0 f, I n ) D d 3 D d 6 n h i + n h i σ i 6 n ) D d h i nν h), 6 σ i f x i+ ) f x i ) h i d + D) ν h) mx h i,,n Proof Applying inequlity 375) on the intervl x i, x i+ for i 0,, n we hve xi+ f t) dt h { ) i xi + x i+ f + f x i ) + f x i+ )} x i 4 h i D d + 6 S i d + D, where S i f x i+) f x i ), h i x i+ x i h i Summing over i for i 0,,, n gives A 0 f, I n ) nd the first bound for R 0 f, I n ) Now, consider the right hnd side of the inequlity bove Then since h i 6 D d + S i d + D S i d + D D d h i D d), 6 Summing over i produces the lst two upper bounds for the error

156 3 THREE POINT QUADRATURE RULES 48 Corollry 35 Let the conditions of Theorem 34 hold Also, let I m be the equidistnt prtition of, b, I m : x i +ih, i 0,,, m with h b m Then f x) dx h 4 f x 0) + f x m ) h m i f x i ) D d 3 b ) m Proof From Theorem 34 with h i b m for ll i nd using the expression for A 0 f, I m ) s given in Corollry 30 produces the desired result Remrk 38 If we wish to pproximte the integrl f t) dt using the bove qudrture rule in Corollry 35, with n ccurcy of ε > 0, then we need m ε N points for the equispced prtition I m, D d b ) m ε +, 3 ε where x denotes the integer prt of x R 35 Inequlities for which the First Derivtive Belongs to L, b In this subsection we discuss the sitution in which f L, b which is liner spce of ll bsolutely integrble functions on, b We use the usul norm nottion, where, we recll, g : g s) ds, g L, b Theorem 36 Let f : I R R be differentible mpping on I the interior of I) nd, b I re such tht b > If f L, b, then the following inequlity holds for ll x, b, α x), x nd β x) x, b, 378) f t) dt β x) α x)) f x) + α x) ) f ) + b β x)) f b) f { b + α x) + x + β x) x + b + x + b + α x) + x β x) b + x } Proof Let K x, t) be s defined in 3) An integrtion by prts produces the identity s given by 33) Thus, from 33), 379) f t) dt β x) α x)) f x) + α x) ) f ) + b β x)) f b) K x, t) f t) dt

157 49 P Cerone nd SS Drgomir Now, using 3), 380) where with nd x αx) K x, t) f t) dt t α x) f t) dt + βx) + x α x) ) α x) t) f t) dt + + β x) x) M x) f The well-known identity my be used to give nd t β x) f t) dt x x β x) t) f t) dt + αx) βx) x αx) βx) f t) dt + x α x)) f t) dt + b β x)) M x) mx {M x), M x)} t α x)) f t) dt t β x)) f t) dt x M x) mx {α x), x α x)} M x) mx {β x) x, b β x)} mx {X, Y } X + Y M x) x M x) b x + X Y αx) + α x) + x + β x) x + b βx) f t) dt f t) dt Thus, using the identity gin gives M x) M x) + M x) + M x) M x) 38) { b + α x) + x + β x) x + b + x + b + α x) + x β x) b + x } On substituting 38) into 380) nd using 379), result 378) is produced nd thus the theorem is proved

158 3 THREE POINT QUADRATURE RULES 50 Corollry 37 Let f stisfy the conditions of Theorem 36 Then α x) +x nd β x) b+x give the best bound for ny x, b nd so 38) f f t) dt b f x) + b + x + b x b ) f ) + b x b ) f b) Proof From 378) the miniml vlue ech of the moduli cn tke is zero Hence the result Remrk 39 An even tighter bound my be obtined from 37) if x is tken to be t the mid-point giving 383) f t) dt b ) + b f ) + f b) f + b f 4 This result corresponds to the verge of mid-point nd trpezoidl qudrture rule for which f L, b Theorem 38 Let f stisfy the conditions s stted in Theorem 336 Then the following inequlity holds for ny γ 0, nd x, b : 384) f t) dt b ) { γ) f x) ) ) x b x + γ f ) + f b)} b b f + γ b + x + b Proof Let α x) nd β x) be s in 3) Then, β x) α x) γ) b ), α x) γ x ), nd Now, b β x) γ b x), γ α x) + x β x) x + b ) x ) γ ) b x) b + α x) + x + β x) x + b b ) + γ

159 5 P Cerone nd SS Drgomir nd x + b + α x) + x β x) x + b x + b + γ x ) γ b x) x + b + γ x + b ) x + b + γ Substitution of the bove results into 378) gives 384), thus proving the theorem Remrk 330 If γ +x in 384) then α x) nd β x) x+b nd so result 38) is rightly recovered The best qudrture rule of this type is given by 383) which is obtined by tking the optiml γ nd x vlues t their respective mid-points of +b nd in 384) Remrk 33 Tking γ 0 in 384) gives Ostrowski s inequlity for f L, b s obtined by Drgomir nd Wng 33 If γ in 384), then generlized trpezoidl rule is obtined for which the best bound occurs when x +b giving the clssicl trpezoidl type rule for functions f L, b Corollry 39 Let f stisfy the conditions s stted in Theorem 36 Then the following inequlity holds for γ 0, : 385) { f t) dt b ) γ) f f b + γ + b ) + γ f ) + f b) } Proof Simply evluting 39) t x +b gives 385) Remrk 33 Tking γ 0 nd into 30) gives the mid-point nd trpezoidl type rules respectively Remrk 333 Tking γ in 385) gives the optiml qudrture rule shown in 383) Plcing γ 3 gives Simpson type rule with n error bound of f b 3 Thus, Simpson type rule is reltively worse by 3 ) when compred with the optiml rule 383) In ddition, the optiml rule is just s esy to implement s the Simpson rule All tht is different re the weights The following results investigte the implementtion of the bove inequlities to numericl integrtion Theorem 330 For ny, b R with < b let f :, b) R be differentible mpping Let f L, b, then, for ny prtition I n : x 0 < x < < x n < x n b of, b nd ny intermedite point vector ξ ξ 0, ξ,, ξ n ) such tht

160 3 THREE POINT QUADRATURE RULES 5 ξ i x i, x i+ for i 0,,, n, we hve, for γ 0,, 386) f x) dx A c f, I n, ξ) f + γ { hi mx 0in + ξ i x } i + x i+ f + γ ν h) where h i x i+ x i, ν h) mx 0in h i nd A c f, I n, ξ) is given by n A c f, I n, ξ) γ) h i f ξ i ) +γ n n ξ i x i ) f x i ) + x i+ ξ i ) f x i+ ) Proof Applying inequlity 384) on the intervl x i, x i+ for i 0,,, n we hve: xi+ f x) dx x i { γ) f ξ i ) h i + γ ξ i x i ) f x i ) + x i+ ξ i ) f x i+ )} + γ hi mx 0in + ξ i x i + x i+ xi+ f x) dx Summing the bove inequlity, we hve 386) Furthermore, observe tht ξ i x i + x i+ h i for i 0,,, n Therefore, hi mx 0in + ξ i x i + x i+ mx h i ν h) 0in nd hence the theorem is proved Remrk 334 The coefficient of the γ term in A c f, I n, ξ) my be simplified to give n n ξ i x i ) f x i ) + x i+ ξ i ) f x i+ ) n n ξ i f x i ) f x i+ ) + x i+ f x i+ ) x i f x i ) n ) ξi ξ i f xi ) + ξ 0 f ) ξ n f b) + bf b) f ) i n ) ξi ξ i f xi ) + ξ 0 ) f ) + ) b ξ n f b) i x i

161 53 P Cerone nd SS Drgomir This version hs the dvntge in tht the number of function evlutions is minimized Thus, 387) n A c f, I n, ξ) γ) h i f ξ i ) + γ { n ) ξi ξ i f xi ) i + ξ 0 ) f ) + b ξ n ) f b) } Corollry 33 Let, b R with < b nd the mpping f :, b) R be differentible Further, let f L, b Then, for ny prtition I n : x 0 < x < < x n < x n b of, b we hve for ny 0 γ, 388) where nd n A f, I n ) γ) h i f f x) dx A f, I n ) + R f, I n ) xi + x i+ R f, I n ) f ) + γ n h i f x i ) + f x i+ ), + γ ν h) Proof The proof is strightforwrd We either strt with Corollry 39 nd follow the procedure of Theorem 330, or we cn tke the esier option of plcing ξ i xi+xi+ in Theorem 330 to immeditely produce the result Remrk 335 The qudrture rule given by 388) is composite mid-point nd trpezoidl rule with γ determining the reltive weighting of the two The optiml rule is obtined when the composition is strightforwrd verge which is obtined by tking γ Corollry 33 Let the conditions of Corollry 33 hold, tking in prticulr γ nd the prtition to be equidistnt so tht I m : x i + ih, i 0,,, m with h b m Then 389) where nd f x) dx A o f, I m ) + R o f, I m ) A o f, I m ) h 4 f ) + f b) + R o f, I m ) f 8 b m m i ) f x i ) Proof From Corollry 33, let subscript of o signify the optiml qudrture rule obtined when γ nd so A o f, I m ) h m ) xi + x i+) f + h m f xi ) + f ) x i+), 4

162 3 THREE POINT QUADRATURE RULES 54 where Now m Thus, f xi ) + f x i+) ) A o f, I m ) h h 4 x i + x i+) m + h i + ) x i+ m m f x 0 ) + f x m ) + f x i ) + f ) x i+) i m f x 0 ) + f x m ) + f x i ) m f x i+ ) + i f x 0 ) + f x m ) + Further, from Corollry 33 with h i b m nd hence the corollry is proved R o f, I m ) f 8 f x i ) m i m i f x i ) + h 4 f x 0) + f x m ) for i 0,,, m nd γ we obtin ) b, Remrk 336 If we wish to pproximte the integrl f t) dt using the qudrture rule A o f, I m ) nd 389) with n ccurcy of ε > 0, then we need m ε N points for the equispced prtition I m where f m ε b ) +, 8 ε where x denotes the integer prt of x R 36 Grüss-type Inequlities for Functions whose First Derivtive Belongs to L, b The identity of Drgomir nd McAndrew 30 s given by 347) will now be utilized to obtin further inequlities Define n opertor σ such tht 390) σ f) f M f), where M f) b f u) du Then, 347) my be written s 39) T f, g) T σ f), σ g)), where T f, g) M fg) M f) M g) It my be noticed from 390) nd 39) tht M σ f)) M σ g)) 0, so tht 39) my be written in the lterntive form: 39) M fg) M f) M g) M σ f) σ g))

163 55 P Cerone nd SS Drgomir Theorem 333 Let f : I R R be differentible mpping on I the interior of I) nd, b I re such tht b > If f L, b, then the following inequlity holds for ll x, b nd γ 0, : 393) f t) dt b ) + b ) γ) x + b ) S σ f ) θ γ, x), where 394) θ γ, x) sup { γ) f x) + γ t,b ) x f ) + b K x, t) γ) x + b ), ) } b x f b) b K x, t) is s given by 35) nd S M f ) with M ) nd σ ) s given by 390) 395) Thus, Proof Applying 39) or 39) on the mppings K x, ) nd f ) gives T K, f ) T σ K), σ f )) M σ K) σ f )) 396) b ) T K, f ) σ f ) sup σ K) t,b Now, 397) θ γ, x) sup σ K) sup K x, t) M K), t,b t,b where, from 33), M K) b K x, u) du γ) x + b ) Further, using identity 30), 396) nd 397), the inequlity 393) is derived nd the theorem is hence proved We now wish to obtin n explicit expression for θ γ, x) s given by 394) Using 35) in 394) gives 398) θ γ, x) sup k x, t), t,b where 399) k x, t) nd φ x), ψ x) re s given by 356) Therefore, { t φ x), t, x t ψ x), t x, b 300) θ γ, x) mx { φ x), x φ x), x ψ x), b ψ x) } since the extremum points from 398) nd 399) re obtined t the ends of the intervls s k x, t) is piecewise liner

164 3 THREE POINT QUADRATURE RULES 56 The representtion 300) my be explicit enough, but it is possible to proceed further, s in Subsection 34, by mking the trnsformtion 30) x δb + δ), δ 0, Using 358) 36) gives 30) θ γ, x) Θ γ, δ) { b ) mx δ γ δ), γ δ), γδ }, δ + γ δ) Now, the expressions in 30) cn be either positive or negtive depending on the region A, B,, E s defined by 369) nd depicted in Figure 3 The well known result mx {X, Y } X + Y + X Y my be pplied twice to give 303) mx {X, Y, Z, W } X + Y + Z + W + X Y + Z W + X + Y ) Z + W ) + X Y Z W Tking heed of Remrk 36 then, since we re now deling with the mximum in 30), tht is, point, then it is possible to investigte the regions B nd E B for γ, δ where B : δ > γ nd E B : δ < γ In region B, from 30), 304) Θ B γ, δ) b ) mx { γ) δ + γ ), γδ γ ), γδ, } γ) δ nd ssociting these elements in order with those of 303) gives X + Y δ, X Y γ ) δ) Z + W γ ) δ, Z W δ) Thus, fter some simplifiction, Θ B 305) b γ, δ) γ + γ) δ ) Similrly, in region E B Θ EB γ, δ) b ) mx { γ) δ + γ, γδ γ ), γδ, } γ) δ nd gin ssociting these elements in order with those of 303) gives X + Y δ, X Y γ ) δ) Z + W δ, Z W γ) δ

165 57 P Cerone nd SS Drgomir Therefore, fter some simplifiction, we obtin Θ EB γ, δ) 306) γ δ + γ) b δ Now 307) Θ A γ, δ) Θ B γ, δ), Θ C γ, δ) Θ B γ, δ) nd Θ D γ, δ) Θ B γ, δ) Let E E E where E represents the region of E for which γ nd E represents the region of E for which γ > Tht is, E E A E D nd E E B E C where E k is the reminder of the squre region contining region k A, B, C, D Hence, using 305) 307) in 30) gives γ + γ ) δ γ + γ) ) δ on A, on B, 308) Θ γ, δ) b γ + γ) δ) on C, γ + γ δ) on D, γ + γ δ on E, γ + γ) δ on E Remrk 337 It my be noticed tht 308) my be simplified to give γ Θ γ, δ) 309) b + γ δ, γ γ + γ) δ, γ nd so, using the fct tht b ) γ ) x +b in 309) gives, b γ) + γ x +b, γ 30) θ γ, x) b γ + γ) x +b, γ Thus the bound in Theorem 333, nmely 394), is explicitly given by 30) Remrk 338 Tking different vlues if γ will produce bounds for vrious inequlities For γ 0 in 393) nd 30), perturbed Ostrowski type inequlity is obtined with uniform bound Nmely, f t) dt b ) f x) x + b ) S b σ f )

166 3 THREE POINT QUADRATURE RULES 58 For γ in 393) nd 30) generlized perturbed trpezoidl rule is obtined with the sme uniform bound viz ) ) x b x f t) dt b ) f ) + f b) x + b ) S b b b σ f ) Corollry 334 Let the conditions on f be s in Theorem 333 Then the following inequlity holds for ny x, b b f t) dt 3) b ) f x) + x ) f ) + b x) f b) b + x + b σ f ) Proof Letting γ in 393) nd 30) redily produces the result Corollry 335 Let the conditions on f be s in Theorem 333 Then the following inequlity holds for ny γ 0, { ) + b f t) dt b ) γ) f + γ } 3) f ) + f b) b σ f γ), γ ) b γ, γ Proof Tking x +b in 393) nd 30) gives the result s stted Remrk 339 Tking x +b in 3) or γ in 3) is equivlent to tking both these in 393) nd 30) This produces the shrpest bound in this cse, giving f t) dt b { ) + b f + } f ) + f b) b σ f ) 4 A Simpson type rule is obtined from 3) if γ 3 is tken, giving bound consisting of b 3 rther thn the b 4 obtined bove A perturbed generlized Simpson type rule my be demonstrted directly from 393) nd its bound from 30) by tking γ 3 to give f t) dt b { ) x f ) + f ) 3 b ) b x + f b) x + b ) S} b b + 3 x + b σ f )

167 59 P Cerone nd SS Drgomir Remrk 340 The numericl implementtion of the inequlities obtined in the current subsection will not be followed up since they follow those of Subsection 35 It my be noticed tht Corollries 334 nd 335 re similr to Corollries 37 nd 39 with σ f ) replcing f In similr fshion, the implementtion of the verge of the mid-point nd trpezoidl rules s developed in Corollry 33 my similrly be developed here with σ f ) replcing f Ech of these norms my be better for differing functions f 37 Inequlities for which the First Derivtive Belongs to L p, b Theorem 336 Let f :, b R be differentible mpping on, b) nd f L p, b) where p > nd p + q Then the following inequlity holds for ll x, b, α x), x nd β x) x, b, 33) f t) dt β x) α x)) f x) + α x) ) f ) + b β x)) f b) α x) ) q+ + x α x)) q+ + β x) x) q+ + b β x)) q+ q q + ) q f p x ) q+ + b x) q+ q f q + p ) b q b ) f q + p where f p : ) b f t) p p dt Proof Let K x, t) be s defined by 3) Then n integrtion by prts of K x, t) f t) dt produces the identity s given by 33) Thus, from 33) : 34) f t) dt β x) α x)) f x) + α x) ) f ) + b β x)) f b) K x, t) f t) dt Now, by Hölder s inequlity we hve: 35) K x, t) f t) dt K x, t) q dt ) q f p

168 3 THREE POINT QUADRATURE RULES 60 Now, from 3), Therefore, 36) q + ) αx) K x, t) q dt βx) + x αx) βx) + x x t α x) q dt + t β x) q dt + αx) x α x) t) q dt + K x, t) q dt β x) t) q dt + βx) αx) t α x) q dt βx) t β x) q dt t α x)) q dt t β x)) q dt α x) ) q+ + x α x)) q+ + β x) x) q+ + b β x)) q+ Thus, using 34), 35) nd 36) gives the first inequlity in 33) Now, using the inequlity 37) z x) n + y z) n y x) n with z x, y nd n >, in 36) twice, tking z α x) nd then z β x), we hve 38) 39) q + ) K x, t) q dt x ) q+ + b x) q+ b ) q+ upon using 37) once more Hence, by utilizing 34), 35) with 38) nd 39) we obtin the second bound nd the third inequlity in 33) Corollry 337 Let the conditions on f of Theorem 336 hold Then α x) +x nd β x) b+x give the best bound for ny x, b nd so 30) f t) dt b f x) + x ) q+ + b x) q+ q + x b q f p ) f ) + b x b ) f b) Proof The inequlity 37) produces n upper bound obtined with z x or y For z x, y nd n > ) n 3) z x) n + y z) n y x

169 6 P Cerone nd SS Drgomir where the lower bound is relized when z x+y Thus tighter bound thn the first inequlity in 33) is obtined when, from 36) nd using 3), α x) +x nd β x) x+b Hence, 30) is obtined nd the corollry is proved Remrk 34 The best inequlity we my obtin from 30) results from utilizing 3) gin, giving, with x +b, f t) dt b { ) + b f + } 3) f ) + f b) ) b ) b q f 4 q + p Motivted by Theorem 336 nd Corollry 337 we now tke α x) nd β x) to be convex combintions of the end points so tht they re s defined in 3) The following theorem then holds Theorem 338 Let f stisfy the conditions s stted in Theorem 336 Then the following inequlity holds for ny γ 0, nd x, b : { ) ) x b x f t) dt b ) γ) f x) + γ f ) + f b)} 33) b b γ q+ + γ) q+ q x ) q+ + b x) q+ q q + ) q f p nd Proof Using α x) nd β x) s defined in 3), then β x) α x) γ) b ), α x) γ x ), b β x) γ b x), x α x) γ) x ) β x) x γ) b x) Substituting these results into the first inequlity of Theorem 336 gives the stted result Remrk 34 Tking γ 0 or in 33) produces the corser upper bound s obtined in the second inequlity of Theorem 336 In ddition, tking x or b in 33) gives the even corser bound s given by the third inequlity of Theorem 336 Here we re utilizing 37) where the upper bound is ttined t z x or y, the end points Corollry 339 Let f stisfy the conditions of Theorems 336 nd 338 Then, the following inequlity holds for ny γ 0, : { ) + b f t) dt b ) γ) f + γ } 34) f ) + f b) γ q+ + γ) q+ ) ) q b b q f q + p

170 3 THREE POINT QUADRATURE RULES 6 Proof From identity 3) the minimum is obtined t the mid-point Therefore, from 33), inf x ) q+ + b x) q+ ) q+ b, x,b when x +b Hence the result 34) Remrk 343 Corollry 337 is recptured if 33) is evluted t γ, the mid-point Remrk 344 Tking γ 0 in 33) produces n Ostrowski type inequlity for which f L p, b s obtined by Drgomir nd Wng 34 Furthermore, tking x +b gives mid-point rule Remrk 345 Tking γ in 33) produces generlized trpezoidl rule for which the best bound occurs when x +b, giving the stndrd trpezoidl ) rule with bound of b q b q+ This bound is twice s shrp s tht obtined by Drgomir nd Wng 34 since they used n Ostrowski type rule nd obtined results t x, x b nd utilized the tringle inequlity Remrk 346 Tking γ +b nd x in 33) gives the best inequlity s given by 3) Tking γ 3 in 34) produces Simpson type rule with bound on the error of ) + q + q+ ) ) q b b q f 3 q + p Tking γ in 34) gives the optiml rule with bound on the error of ) ) b b q f q + p Thus, there is reltive difference of + q+ 3 3 ) q between Simpson type rule nd the optiml When q for exmple, the reltive difference is The gretest the reltive difference cn be is 3 The following prticulr instnce for Eucliden norms is of interest Corollry 340 Let f :, b R be differentible mpping on, b) nd f L, b) Then the following inequlity holds for ll x, b nd γ 0, : { ) ) x b x f t) dt b ) γ) f x) + γ f ) + f b)} 35) b b ) b γ ) b ) + 3 x + b ) f

171 63 P Cerone nd SS Drgomir Proof Applying Theorem 338 for p q immeditely gives the left hnd side of 35) with bound of 36) γ 3 + γ) 3 x ) 3 + b x) 3 f 3 Now, using identity 34) we get: γ 3 + γ) γ ) nd x ) 3 + b x) 3 b b ) + 3 x + b ), which, upon substitution into 36) gives 35) Remrk 347 The numericl implementtion of the inequlities in this subsection follows long similr lines s treted previously The only difference is in the pproximtion of the bound nd knowledge of f p, which need to be determined priori in order tht the corseness of the prtition my be clculted, given prticulr error tolernce 38 Grüss-type Inequlities for Functions whose First Derivtive Belongs to L p, b From 39) nd 39) we hve 37) T f, g) M σ f) σ g)), where σ f) represents shift of the function by its men, M s given in 390) Thus, using Hölder s inequlity from 37) gives 38) b ) T f, g) σ f) q σ g) p, where nd we sy h L p, b h p : h t) p dt Theorem 34 Let f :, b R be differentible mpping on, b) nd f L p, b) where p > nd p + q The following inequlity then, holds for ll x, b, { ) ) } x b x 39) f t) dt b ) γ) f x) + γ f ) + f b) b b b ) γ) x + b ) S σ K x, )) q σ f ) p, ) p where σ ) is s given in 390) nd K x, ), S re s in 35)

172 3 THREE POINT QUADRATURE RULES 64 Proof Identifying K x, ) with f ) nd f ) with g ) in 38) gives 330) b ) T K x, ), f ) σ K x, )) q σ f ) p Further, using identities 3), 34) nd the fct tht S b b f t) dt in 330) redily produces 39) nd hence the theorem is proved We now wish to obtin closed form expression for σ K x, )) q Notice tht σ K x, t)) K x, t) K x, u) du, b where K x, t) is s given by 35) nd so using 34) { t φ x), t, x 33) K s x, t) σ K x, t)) t ψ x), t x, b where φ nd ψ re s presented in 356), Thus, 33) K s x, t) q dt x t φ x) q dt + Using 33) in 33) gives σ K x, )) q K s x, ) q K s x, t) q dt x t ψ x) q dt nd upon mking the respective substitutions b ) u t φ x) nd b ) v t ψ x) for the integrls on the right hnd side, { b x φx) φx) } 333) K s x, t) q dt b ) q+ b u q b du + v q dv φx) x φx) b b Following the procedure of Subsection 34, we my mke the substitution 334) x δb + δ), to give, from 333), nd using 358) 36), 335) I q) γ, x) b ) q+ J q) γ, δ) b ) q+ J q) γ, δ) + J q) γ, δ), where 336) I q) γ, x) K s x, t) q dt, ) q, 337) J q) γ, δ) w+ δ w v q dv, nd w γδ 338) J q) γ, δ) J q) γ, δ)

173 65 P Cerone nd SS Drgomir Now, from 336), the limits my be both negtive, one negtive nd one positive, or both positive Therefore, with w γδ, 339) q + ) J q) γ, δ) w) q+ δ w) q+, γ) < δ < γ ; w) q+ + w δ) q+, δ < γ, δ < γ) ; w + δ) q+ w q+, γ < δ < γ) Further, from 336) nd 337) 340) q + ) J q) γ, δ) w) q+ δ w) q+, γ) < δ < γ ; w) q+ + w + δ) q+, δ > γ, δ > γ) ; w + δ) q+ w q+, γ < δ < γ), where w γ) δ) We re now in position to combine 339) nd 340) by using the result 335) on ech of the regions A,, E s given by 369) nd depicted in Figure 3 Thus, 34) q + ) J q) γ, δ) w) q+ δ w) q+ + w) q+ + w + δ) q+ on A; w δ) q+ w q+ + w) q+ + w + δ) q+ on B; w) q+ + w + δ) q+ + w) q+ δ w) q+ on C; w) q+ + w δ) q+ + w + δ) q+ w) q+ on D; w) q+ + w + δ) q+ + w) q+ + w + δ) q+ on E Hence σ K x, )) q is explicitly determined from 33), 335) nd 340) on using 334) nd the fct tht w γδ nd w γ) δ) Remrk 348 It is instructive to tke different vlues of γ to obtin vrious inequlities tht led to vriety of qudrture rules For γ 0 then, from Figure 3 it my be seen tht we re on the left boundry of regions A nd D so tht, from 34), uniform bound independent of δ is obtined to give ) q q + ) J q) 0, δ)

174 3 THREE POINT QUADRATURE RULES 66 Using 335), 33) nd 39) produces perturbed Ostrowski inequlity 34) b f t) dt b ) f x) x + b ) S ) b q σ f ) q + p Evlution t x +b gives the mid-point rule In similr fshion, for γ the right hnd boundry of B nd C results to produce perturbed generlized trpezoidl inequlity 343) b f t) dt b ) ) b q σ f ) q + p ) x f ) + b ) b x f b) x + b ) S b Tking x +b produces the trpezoidl rule for which σ f ) L p, b Corollry 34 Let the conditions on f be s in Theorem 34 Then the following inequlity holds for ny x, b 344) b f t) dt b ) f x) + x ) f ) + b x) f b) x ) q+ + b x) q+ q σ f ) q + p b ) b q σ f ) q + p Proof Plcing γ in 34) gives, fter some simplifiction, q + ) J q), δ ) Hence, from 334) nd 335), ) q δ q+ + δ) q+ ) ) q x I q), x ) q+ + b x) q+ q + From 33) nd 336), tking the q th root of the bove expression gives 344) from 39) on tking γ The second inequlity is obtined on using 37)

175 67 P Cerone nd SS Drgomir Corollry 343 Let the conditions on f be s in Theorem 34 Then the following inequlity holds for ny γ 0, { ) + b f t) dt b ) γ) f + γ } 345) f ) + f b) b ) b q γ q+ + γ) q+ q σ f ) q + p b ) b q σ f ) q + p Proof Tking γ in 34) plces us in the region E nd so q + ) J q) γ, ) ) q γ q+ + γ) q+ q nd, from 334) nd 335), I q) γ, + b ) b )q+ q + ) q γ q+ + γ) q+ From 33) nd 336), tking the q th root of the bove expression produces 345) from 39) on tking x +b The second inequlity is esily obtined on using 37) Remrk 349 Tking x +b in 344) or γ in 345) is equivlent to tking both of these in 39) nd using 33), 334) 336) nd q + ) J q), ) ) q 4 from 34) This produces the shrpest inequlity see 33)) in the clss Nmely, f t) dt b { ) + b f + } 346) f ) + f b) b ) b q σ f ) 4 q + p Result 346) my be compred with 3) nd it my be seen tht either one my be better, depending on the behviour of f A Simpson type rule is obtined from 345) if γ 3, giving bound consisting of q times the bove bound for the verge of midpoint nd trpezoidl 3 + q+ 3 rule For the Eucliden norm, q nd so Simpson s rule hs bound of 3 times tht of the verge of the midpoint nd trpezoidl rule A generlized Simpson type rule my be obtined by tking γ 3 in 39) nd using 33), 334) 336) nd 34) in much the sme wy s Remrk 339 Remrk 350 Corollries 34 nd 343 my be implemented in stright forwrd fshion s crried out in erlier subsections The bounds involve determining σ f ) in dvnce to decide on the refinement of the grid tht is required in order to chieve prticulr ccurcy

176 3 THREE POINT QUADRATURE RULES Three Point Inequlities for Mppings of Bounded Vrition, Lipschitzin or Monotonic The following result involving Riemnn-Stieltjes integrl is well known It will be proved here for completeness Lemm 344 Let g, v :, b R be such tht g is continuous on, b nd v is of bounded vrition on, b Then g t) dv t) exists nd is such tht b 347) g t) dv t) sup g t) v), t,b where b v) is the totl vrition of v on, b Proof We only prove the inequlity 347) Let n : < x n) 0 < x n) < < x n) n < xn) n b be sequence of prtitions of, b such tht ν n ) 0 s n where ν n ) : mx i {0,,,n } h n) i ξ n) i x n) i, x n) i+ for i 0,,, n then where g t) dv t) lim n ν n) 0 n lim ν n) 0 g g sup g t) t,b ξ n) i ξ n) i b v), ) v ) v with h n) i x n) i+ x n) i+ x n) ) v ) v i+ xn) i x n) i x n) i ) ) Let 348) b v) sup n n v x n) i+ ) v x n) i ), nd n is ny prtition of, b Theorem 345 Let f :, b R be mpping of bounded vrition on, b Then the following inequlity holds 349) f t) dt β x) α x)) f x) + α x) ) f ) + b β x)) f b) b f) { b + α x) + x + β x) x + b + b + x + β x) x + b α x) + x }, where α x), x nd β x) x, b

177 69 P Cerone nd SS Drgomir Proof Let the Peno kernel be s defined in 3), then consider the Riemnn- Stieltjes integrl K x, t) df t) giving K x, t) df t) x t α x)) df t) + x t α x)) f t) x t f t) dt x t β x)) df t) + t β x)) f t) b tx f t) dt Simplifying nd grouping some of the terms together produces the identity 350) K x, t) df t) β x) α x) f x) + α x) f ) + b β x) f b) Now, to obtin the bounds from our identity 350), x K x, t) df t) t α x)) df t) + x t α x)) df t) + x x f t) dt t β x)) df t) t β x)) df t) Further, using the result of Lemm 344, nmely 347) on ech of the intervls, x nd x, b by ssociting g t) with t α x) nd t β x) respectively gives, on tking dv t) df t), 35) K x, t) df t) Let nd so sup t α x) t,x b m x) f) x f) + sup t β x) t x,b x b f) m x) sup t α x) mx {α x), x α x)} t,x 35) m x) x Similrly, nd so + α x) + x m x) sup t β x) mx {β x) b, b β x)} t x,b 353) m x) b x + β x) x + b x

178 3 THREE POINT QUADRATURE RULES 70 Thus, from 35) 354) K x, t) df t) where Therefore, x b m x) f) + m x) f) m x) b f), m x) mx {m x), m x)} 355) m x) m x) + m x) + m x) m x) Substitution of m x) nd m x) from 35) nd 353) into 354) nd using 350) gives inequlity 349), nd the theorem is proved It should be noted tht it is now possible to tke vrious α x) nd β x) to obtin the previous results For exmple, tking α x) β x) x produces the results of Drgomir, Cerone nd Perce 5 involving the generlized trpezoidl rule Further, evlution t x +b of this result gives the clssicl trpezoidl type rule s obtined by Drgomir 0 Tking α x) nd β x) b reproduces the Ostrowski rule for functions of bounded vrition 9 In prticulr, we shll tke α x) nd β x) to be convex combintions of the end points to obtin the following theorem Theorem 346 Let f stisfy the conditions of Theorem 345 Then the following inequlity holds for ny γ 0, nd x, b : 356) f t) dt { ) ) x b x b ) γ) f x) + γ f ) + f b)} b b + γ b + x + b b f) nd Proof Let α x), β x) be s in 3) Then, from Theorem 345 β x) α x) γ) b ), α x) γ x ), b β x) γ b x) Further, from 35), 353), nd 355), m x) + γ ) x ) m x) + γ ) b x) x

179 7 P Cerone nd SS Drgomir nd m x) + γ b + x + b Substitution of m x) into 354), nd using the identity 350) gives the result 356) nd the theorem is thus proved Remrk 35 We note tht corser uniform bounds my be obtined on using the fct tht mx X A,B X A + B B A Remrk 35 A tighter bound is obtined when min X A,B X A + B The minimum of 0 is ttined when X A+B Corollry 347 Let f stisfy the conditions of Theorems 345 nd 346 Then the following inequlity holds for ny γ 0, : 357) b f t) dt b ) { γ) f + γ b f) + b ) + γ f ) + f b) } Proof From 354) nd Remrk 35, min x,b x + b 0 when x +b Hence the result 355) is obtined Corollry 348 Let f stisfy the conditions of Theorems 345 nd 346 Then the following inequlity holds for ll x, b, b f t) dt 358) {b ) f x) + x ) f ) + b x) f b)} b + x + b b f) Proof From 356) nd Remrk 35, min γ 0, γ 0, when γ Thus, plcing γ in 356) gives the result 358) Remrk 353 The shrpest bounds on 357) nd 358) occur when γ nd x +b s my be concluded from the result of Remrk 35 The sme result

180 3 THREE POINT QUADRATURE RULES 7 cn be obtined directly from 356), giving the qudrture rule with the shrpest bound s 359) f t) dt b ) + b f ) + f b) f + b b f) 4 It should be noted, s previously on similr occsions, tht tking γ 3 in 357) produces Simpson-type rule s obtined by Drgomir 8 which is worse thn the optiml 3 point Lobtto rule s given by 359) Nmely, 360) f t) dt b ) + b f ) + f b) f + 3 b b f), 3 which is worse thn 359) by n bsolute mount of Computtionlly speking, the Simpson type rule 360) is just s efficient nd esy to pply s the optiml rule 359) which is the verge of trpezoidl nd mid- point rule Remrk 354 Tking vrious vlues of γ 0, nd/or x, b will reproduce erlier results Tking γ 0 in 356) will reproduce the results of Drgomir 9, giving n Ostrowski integrl inequlity for mppings of bounded vrition In ddition, tking x +b would give mid-point rule If γ is substituted into 357), then the results of Drgomir, Cerone nd Perce 5 re recovered, giving generlized trpezoidl inequlity for ny x, b Furthermore, fixing x t its optiml vlue of +b would give the results of Drgomir 9 Putting γ 3 in 356), we obtin f t) dt 3 b ) f x) + x ) f ) + b x) f b) b + 3 x + b b f), which is generlized Simpson type rule Also, tking x +b gives the result 360), which ws lso produced in Drgomir 8 Remrk 355 If f is bsolutely continuous on, b nd f L, b, then f is of bounded vrition By pplying the theorems of this subsection, the theorems of Subsection 35 re hence recovered Thus, replcing b f) by f in this subsection reproduces the results of Subsection 35 nd vice vers, provided tht the conditions on f re stisfied Further, the perturbed three point qudrture rules obtined in Subsection 36 through Grüss-type inequlities my be obtined here, where, insted of σ f ) in identity 393), we would hve b σ f)) b f) Thus the following theorem would result

181 73 P Cerone nd SS Drgomir Theorem 349 Let f :, b R be mpping of bounded vrition on, b Then the following inequlity holds 36) f t) dt { ) ) } x b x b ) γ) f x) + γ f ) + f b) b b + b ) γ) x + b ) S b θ γ, x) f), where θ γ, x) is s given by 30) nd S is the secnt slope Proof Identifying σ K x, )) with g ) nd σ f )) with v ) in 347) gives, upon noting tht b σ f)) b f), nd dσ f) df since the σ opertor merely shifts function by its men, b 36) σ K x, t)) df t) sup σ K x, t)) f), where σ K x, t)) nd φ x), ψ x) re s given in 356) t,b { t φ x), t, x, t ψ x), t x, b The Riemnn-Stieltjes integrl my be integrted by prts to produce n identity similr to 350) with α x) nd β x) replced by φ x) nd ψ x) respectively, since we re now considering σ K x, )) rther thn K x, ) In other words, 363) σ K x, t)) df t) ψ x) φ x) f x) + φ x) f ) + b ψ x) f b) which becomes, on using 356), 364) σ K x, t)) df t) b ) γ) f x) + γ b x) + x + b + γ x ) x + b ) f b) f t) dt ) f ) f t) dt, A strightforwrd reorgniztion of 363), on noting tht S fb) f) b nd using 36) redily produces 36) where θ γ, x) sup t,b σ K x, t)), nd hence the theorem is proved

182 3 THREE POINT QUADRATURE RULES 74 Remrk 356 Identity 363) or indeed 350)) demonstrtes tht three point qudrture rule my be obtined for rbitrry functions φ ) nd ψ ) or α ) nd β )) Definition The mpping u :, b R is sid to be L Lipschitzin on, b if 365) u x) u y) L x y for ll x, y, b The following lemm holds Lemm 350 Let g, v :, b R be such tht g is Riemnn integrble on, b nd v is L-Lipschitzin on, b Then 366) g t) dv t) L g t) dt Proof Let n : < x n) 0 < x n) < < x n) n < xn) n b be sequence of prtitions of, b such tht ν n ) 0 s n, where ν n ) : mx i {0,,,n } h n) i with h n) i x n) i+ xn) i Further, let ξ n) i x n) i, x n) i+ such tht n ) ) ) g t) dv t) lim g ξ n) i v x n) i+ v x n) i ν n) 0 ) ) n ) lim g ξ n) v x n) i+ v x n) i ) i ν n) 0 x n) i+ x n) xn) i+ i i n ) ) L lim g ξ n) i x n) i+ ν i n) 0 L Hence the lemm is proved g t) dt Theorem 35 Let f :, b R be L-Lipschitzin on, b Then the following inequlity holds 367) f t) dt β x) α x)) f x) + α x) ) f ) + b β x)) f b) { b ) L + x + b ) + α x) + x ) + β x) x + b ) }, where α x), β x) re s given by 3)

183 75 P Cerone nd SS Drgomir Proof The proof is strightforwrd from identity 348), giving, fter tking the bsolute vlue K x, t) df t) L K x, t) dt, since f is L-Lipschitzin, nd thus Lemm 350 my be used Now, K x, t) is s given by 3) nd K x, t) dt Q x) given in 36) Using identity 35) simplifies the expression for Q x) in 36) to give result 367) Hence the theorem is proved Remrk 357 If f is L-Lipschitzin on, b, then the bound on the Riemnn- Stieltjes integrl 368) K x, t) df t) L K x, t) dt On the other hnd, if f is differentible on, b nd f L, b, then the Riemnn integrl K x, t) f t) dt f K x, t) dt Thus, ll the theorems nd bounds obtined in Subsection 3 re pplicble here if f is L-Lipschitzin The f norm is simply replced by L Theorem 35 Let f :, b R be L Lipschitzin on, b Then 369) f t) dt ψ x) φ x)) f x) + φ x) ) f ) + b ψ x)) f b) where L σ K x, )), Proof Consider σ K x, t)) { t φ x), t, x, t ψ x), t x, b σ K x, t)) df t), giving identity 363) nd using 366) redily produces result 369) Thus, the theorem is proved Remrk 358 If φ x) nd ψ x) re tken s in 356), then { ) x 370) f t) dt b ) γ) f x) + γ f ) b ) } b x + f b) + b ) γ) x + b ) S b L I γ, x), where I γ, x) σ K x, )) J γ, x ) b

184 which is given in 370) nd S fb) f) b 3 THREE POINT QUADRATURE RULES 76 Lemm 353 Let g, v, b R be such tht g is Riemnn integrble on, b nd v is monotonic nondecresing on, b Then 37) g t) dv t) g t) dv t) Proof Let n : < x n) 0 < x n) < < x n) n < xn) n b be sequence of prtitions of, b such tht ν n ) 0 s n where ν n ) : mx i {0,,,n } h n) i with h n) i x n) i+ xn) i Now, let ξ n) i x n) i, x n) i+ so tht n ) ) ) g t) dv t) lim g ξ n) i v x n) i+ v x n) i ν n) 0 g n lim ν n) 0 ξ n) i ) v x n) i+ Now, using the fct tht v is monotonic nondecresing, then n g t) dv t) lim ) g ξ n) i v x n) i+ ν n) 0 ) v ) v Mking use of the definition of the integrl, the lemm is proved x n) i x n) i ) )) Theorem 354 Let f :, b R be monotonic nondecresing mpping on, b Then the following inequlity holds: 37) f t) dt β x) α x)) f x) + α x) ) f ) + b β x)) f b) x α x) + β x)) f x) + b β x)) f b) α x) ) f ) sgn K x, t)) f t) dt 373) x α x) + β x)) f x) + b β x)) f b) α x) ) f ) + α x) + x) f α x)) + β x) x + b) f β x)), where K x, t) is s given by 3) nd α x), x, β x) x, b Proof Let the Peno kernel K x, t) be s given by 3) Then the identity 350) is obtined upon integrtion by prts of the Riemnn-Stieltjes integrl K x, t) df t) Now, since f is monotonic nondecresing, then, using Lemm 353 nd identifying f ) with v ) nd K x, ) with g ) in 37) gives: 374) K x, t) df t) K x, t) df t)

185 77 P Cerone nd SS Drgomir Now, from 3), K x, t) df t) x αx) t α x) df t) + βx) + x x α x) t) df t) + x β x) t) df t) + Integrtion by prts nd some grouping of terms gives 375) K x, t) df t) t β x) df t) αx) βx) t α x)) df t) t β x)) df t) x α x) + β x)) f x) + b β x)) f b) α x) ) f ) { αx) x βx) } b + f t) dt f t) dt + f t) dt f t) dt αx) Using the fct tht the terms in the brces re equl to sgn K x, t)) f t) dt, where if u > 0 sgn x,b u x), if u < 0 then, from identity 350) nd eqution 374) we obtin 37) nd thus, the first prt of the theorem is proved Now for the second prt Since f ) is monotonic nondecresing, αx) x αx) f t) dt α x) ) f α x)), f t) dt x α x)) f α x)), x βx) nd Thus, from 375) βx) x βx) f t) dt β x) b) f β x)), f t) dt b β x)) f β x)) 376) K x, t) df t) x α x) + β x)) f x) + b β x)) f b) α x) ) f ) + α x) + x) f α x)) + β x) x + b) f β x)) Substituting 376) into 374) nd utilizing 350), we obtin 373) Thus, the second prt of the theorem is proved

186 3 THREE POINT QUADRATURE RULES 78 Remrk 359 It is now possible to recpture previous results for monotonic nondecresing mppings If α x) β x) x, then the result obtined by Drgomir, Cerone nd Perce 5 for the generlized trpezoidl rule is recovered Moreover, tking x +b gives the trpezoidl-type rule Tking α x) nd β x) b reproduces n Ostrowski inequlity for monotonic nondecresing mppings which ws developed by Drgomir Tking α x) +x nd β x) x+b gives from 373) : b f t) dt b ) f x) + x ) f ) + b x) f b) x + b ) f x) + b x) f b) x ) f ), which is the Lobtto type rule obtined by Milovnović nd Pečrić 45, p 470 However, here it is for monotonic functions As discussed erlier, it is much more enlightening to tke α x) nd β x) to be liner combintion of the end points, nd so the following theorem cn be shown to hold Theorem 355 Let f :, b R be monotonic nondecresing mpping on, b Then the following inequlity exists 377) f t) dt { ) ) x b x b ) γ) f x) + γ f ) + f b)} b b γ) x + b ) f x) + γ b x) f b) x ) f ) 378) 379) sgn K x, t)) f t) dt x ) { γ) f x) f α x)) + γ f α x)) f )} + b x) { γ) f β x)) f x) + γ f b) f β x))} + γ b + x + b f b) f )), where K x, t) is s given by 3) nd α x), β x) by 3) Proof Let α x), β x) be s in 3) Then, from Theorem 354, In ddition, β x) α x) γ) b ), α x) γ x ) nd b β x) γ b x) x α x) + β x)) γ) x + b ),

187 79 P Cerone nd SS Drgomir nd so using these results in 37), 377) is obtined nd the first prt is proved Now for the second prt Note tht nd α x) + x) γ ) x ) β x) x + b) γ) b x) Substituting the bove expressions into the right hnd side of 373) gives γ) x + b ) f x) + γ b x) f b) x ) f ) + γ ) x ) f α x)) + γ) b x) f β x)), which, upon rerrngement, produces 378) Now, to prove 379), the well known result for the mximum my be used, nmely mx {X, Y } X + Y + X Y Thus, from the right hnd side of 378), using mx {γ, γ)} + γ, gives + γ {x ) f x) f )) + b x) f b) f x))} Furthermore, using mx {x, b x} b + x + b redily produces 379) where f b) > f ), since f is monotonic nondecresing Hence the theorem is completely proved Remrk 360 Tking α x) nd β x) to be convex combintion of the endpoints produces, it is rgued, more elegnt bound 379) thn would otherwise be the cse The right hnd side of 376) my esily be shown to equl α x) ) f α x)) f ) + x α x)) f x) f α x)) + β x) x) f β x)) f x) + b β x)) f b) f β x)) M x) f b) f ) where which is s given in 38) M x) mx {α x), x α x), β x) x, b β x)} It is further rgued tht the product form bound in 377) 379) when convex combintion of the end points for α x) nd β x) is tken, it is much more enlightening thn the bound given by 37) nd 373)

188 3 THREE POINT QUADRATURE RULES 80 Corollry 356 Let f stisfy the conditions s stted in Theorem 355 Then the following inequlities hold for ny x, b : b f t) dt b ) f x) + x ) f ) + b x) f b) 380) 38) x ) b x) f x) f ) + f b) f x) b + x + b f b) f )) Proof Plcing γ 38) in 378) nd 379) redily produces 380) nd Corollry 357 Let f stisfy the conditions of Theorem 355 Then the following inequlities hold for ll γ 0, { ) + b f t) dt b ) γ) f + γ } f ) + f b) 38) 383) b {γ f b) f ) )) ))} + b + b + γ) f β f α b ) + γ f b) f )) Proof Tking x +b into 378) redily produces 38) fter some minor simplifiction Plcing x +b into 379) gives 383) Remrk 36 The monotonicity properties of f ) my be used to obtin bounds from 378) or indeed, 380) nd 38)) Now, since f is monotonic nondecresing, then f ) f α x)) f x) f β x)) f b), for ny x, b nd α x), x, β x) x, b Hence, the right hnd side of 378) is bounded by x ) f x) f ) + b x) f b) f x), for ny γ 0, tht is uniform bound) This is further bounded by b + x + b f b) f )), upon using the mximum identity, which is the corsest bound possible from 379), nd is obtined by only controlling the γ prmeter Remrk 36 Although 377), 378) nd its prticulriztions 380) nd 38) re of cdemic interest, their prcticl pplicbility in numericl qudrture is computtionlly restrictive Hence, the bound 379) nd its speciliztions 38) nd 383) re emphsized

189 8 P Cerone nd SS Drgomir Remrk 363 In the foregoing work we hve ssumed tht f is monotonic nondecresing If f is ssumed to be simply monotonic, then the modulus sign is required for function differences Thus, for exmple, in 379), f b) f ) would be required rther thn simply f b) f ) Alterntively, if f ) were monotoniclly nondecresing, then f ) would be monotoniclly nonincresing Remrk 364 Tking vrious vlues of γ 0, nd/or x, b will produce some specific specil cses Plcing γ 0 in 379) gives generlized trpezoidl rule for monotonic mppings nd the results of Drgomir, Cerone nd Perce 5 re recovered If x +b, then the trpezoidl rule results Remrk 365 The optimum result from inequlity 379) is obtined when both γ nd x re tken to be t the midpoints of their respective intervls Thus, the best qudrture rule is: 384) b 4 f t) dt f b) f )) b ) f + b ) + f ) + f b) This result could equivlently be obtined by tking γ +b in 383) or x in 38) Tking γ 3 in 379) gives generlized Simpson-type rule in which the interior point is unspecified Nmely, b f t) dt 3 b ) f x) + x ) f ) + b x) f b) b + 3 x + b f b) f )) If x is tken t the midpoint, then the Simpson-type rule is obtined viz, 385) b 3 f t) dt b ) 3 f b) f )), f + b ) + f ) + f b)) which is worse bound thn 384) Computtionlly, there is no difference in implementing 384) or 385), nd yet 385) is worse by n bsolute mount of

190 3 THREE POINT QUADRATURE RULES 8 Theorem 358 Let f :, b R be monotonic non-decresing mpping on, b Then the following inequlity holds 386) 387) 388) { b f t) dt b ) γ) f x) + γ ) } b x + f b) + b ) γ) x + b b γ x + b ) f x) + γ x ) x + b γ b x) + x + b ) f ) γ sgn σ K x, t))) f t) dt x + b ) f x) + γ x ) γ b x) + x + b ) x f ) b ) S ) f b) x + b ) f b) ) f ) + γ ) b x) f φ x)) + γ) x ) f ψ x)) b + x + b {γ f b) f )) + γ) f ψ) f φ))} Proof From 354), nd identifying f ) with v ) nd σ K x, )) with g ) gives 389) σ K x, t)) df t) σ K x, t)) df t), which is equivlent to 374) with σ K x, t)) replcing K x, t) The results of Theorem 354 re obtined, nmely, by 358) nd 373) with φ ), ψ ) nd σ K x, )) replcing α ), β ) nd K x, ) respectively Tking φ x) nd ψ x) s given by 356) then gives ψ x) φ x) γ) b ), φ x) γ b x) + x + b ), b ψ x) γ x ) x + b ), x φ x) + ψ x)) γ x + b )

191 83 P Cerone nd SS Drgomir Thus, from 37), with the pproprite chnges to φ, ψ nd σ K), we hve γ) b ) f x) + γ b x) + x + b ) 390) f ) + γ x ) x + b ) f b) γ) b ) f x) + γ x ) f ) + b x) f b) b ) γ) x + b ) S nd 39) x φ x) + ψ x)) f x) + b ψ x)) f b) φ x) ) f ) γ x + b ) f x) + γ x ) x + b ) f b) γ b x) + x + b ) f ) Hence, combining 390) nd 39) redily gives the first inequlity Now, for the second inequlity, we hve from 356), φ x) + x) γ ) b x) nd ψ x) x + b) γ) x ) Thus, from 373) nd identifying α with φ nd β with ψ redily gives the second inequlity of the theorem The third inequlity 388) is obtined by grouping the terms in 387) s the coefficients of x nd b x, nd then using the fct tht mx {x, b x} b + x + b Thus, the theorem is now completely proved Remrk 366 From 389), we hve tht σ K x, t)) df t) σ K x, t)) df t) sup σ K x, t)) t,b θ γ, x) f b) f )), df t) where θ γ, x) is s given by 30) Thus, result 36) is obtined becuse, for monotonic nondecresing functions, b f) f b) f ) Remrk 367 Applictions in probbility theory re worthy of mention For X rndom vrible tking on vlues in the finite intervl, b, the cumultive distribution function F x) is defined by F x) Pr X x) x f t) dt Thus, from 356), 369) with 3), we obtin rules for evluting the cumultive

192 3 THREE POINT QUADRATURE RULES 84 distribution function in terms of function evlution of the density Nmely, for ny y, x F x) { γ) x ) f y) + γ y ) f ) + x y) f x)} + γ x + y +x x f) L 4 + ) γ x ) ) + y +x The bove results could be used to pproximte Pr c X d) where c, d, b If γ 0, then the results of Brnett nd Drgomir re recptured If f t) F t), then F t) dt b E X nd so F t) is monotonic nondecresing In ddition, F ) 0, F b) give, from Theorem 358, β x) E X γ) b ) F x) γ) x + b ) F x) + γ b x) x ) γ) F x) + γ ) F α x)) + b x) γ ) F x) + γ) F β x)) + γ γ b + x + b, sgn K x, t)) f t) dt where α x) nd β x) re s given in 3) nd K x, t) by 3) Noting tht R x) Pr {X x} F x), then bounds for α x) E X + γ) b ) R x) could be obtined from the bove development This is more suitble for work in relibility where f x) is filure density Tking vrious vlues of γ 0, nd x, b gives vriety of results, some of which γ 0) hve been obtined in Brnett nd Drgomir 30 Conclusion nd Discussion The work of this section hs investigted three-point qudrture rules in which, t most, the first derivtive is involved The mjor thrust of the work ims t providing priori error bounds so tht suitble prtition my be determined tht will provide n pproximtion which is within prticulr specified tolernce The work contins, s specil cses, both open nd closed Newton-Cotes formule such s the mid-point, trpezoidl nd Simpson rules The results minly involve Ostrowski-type rules which contin n rbitrry point x, b These rules my be utilised when dt is only known t discrete points, which my be non- uniform, without first interpolting The pproch tken hs been through the use of pproprite Peno kernels, resulting in n identity The identity is then exploited through the Theory of Inequlities to obtin bounds on the error, subject to vriety of norms The results developed in the current work provide both Riemnn nd Riemnn-Stieltjes qudrture rules Grüss-type results re obtined, giving perturbed qudrture rules A premture Grüss pproch hs produced rules tht hve tighter bounds A new identity introduced recently by Drgomir nd McAndrew 30 is exploited in Subsections 34,

193 85 P Cerone nd SS Drgomir 36, 38 nd within Subsection 39 to produce Ostrowski-Grüss type results tht seem like perturbtion of the originl three-point qudrture rule A simple reorgnistion of the rule to incorporte the perturbtion produces different three- point rule In effect, the following identity holds, 39) T f, g) T σ f), g) T f, σ g)) T σ f), σ g)), where nd Tht is, T f, g) M fg) M f) M g), σ f) f M f) M f) b f u) du 393) M fg) M f) M g) M σ f) g) M fσ g)) M σ f) σ g)), since M σ f)) M σ g)) 0 Reltion 39) or indeed 393)) does not seem to hve been fully relised in the literture Drgomir nd McAndrew 30 used only the equlity of the outside two terms in 39) or equivlently 393)) to obtin results for trpezoidl rule As mtter of fct, there re only two rules tht hve been used in the current rticle in which f ) K x, ) nd g ) f ) For K x, t) s given by 3), identity 33) is obtined A similr identity to 33) would be obtined if σ K x, )) were to be considered with α x) nd β x) being replced by φ x) nd ψ x) respectively, where { t φ x), t, x 394) σ K x, t)) t ψ x), t x, b Hence, from 3), where Therefore, M K x, )) σ K x, t)) K x, t) M K x, )), K x, u) du b x u α x)) du + b u β x)) du b b x ) ) ) ) + x x x + b b x α x) + β x) b b + b ) ) x b x α x) β x) b b 395) φ x) α x) + M K x, )) + b + α x) β x)) ) b x b

194 3 THREE POINT QUADRATURE RULES 86 with 396) ψ x) β x) + M K x, )) + b + β x) α x)) ) x b Thus, from 33) nd 394) we hve, on using 395) nd 396) nd identifying φ ) with α ) nd ψ ) with β ), σ K x, t)) f t) dt ψ x) φ x)) f x) + φ x) ) f ) + b ψ x)) f b) β x) α x)) f x) + α x) ) f ) + b β x)) f b) +M K x, )) f b) f ) f t) dt f t) dt Therefore, using T K x, ), f ), giving rise to wht seems to be perturbed qudrture rule, is equivlent to considering Peno kernel shifted by its men Nmely, T σ K x, )), f ) M σ K x, )), f ) Bounds on the bove qudrture rule my be obtined using vriety of norms s shown in the section Using the bove identity, bounds involving the first derivtive result If in 39) or 393), σ f ) is used rther thn f, then bounds involving the norms of σ f ) would result Either choice my be superior depending on the prticulr function f ) For the Riemnn-Stieltjes integrl df dσ f) nd so the two cses re equivlent The work of this section hs provided mens for estimting the prtition required in order to be gurnteed certin ccurcy for Newton- Cotes qudrture rules The efficiency, minly in terms of the number of function evlutions to chieve prticulr ccurcy, is very importnt prcticl considertion 33 Bounds for n Time Differentible Functions 33 Introduction Recently, Cerone nd Drgomir 5 see lso Section 3, 350)) obtined the following three point identity for f :, b, b nd α :, b, b, β :, b, b, α x) x, β x) x, then 397) f t) dt β x) α x)) f x) + α x) ) f ) + b β x)) f b) K x, t) df t), where t α x), t, x 398) K x, t) t β x), t x, b They obtined vriety of inequlities for f stisfying different conditions such s bounded vrition, Lipschitzin or monotonic For f bsolutely continuous then

195 87 P Cerone nd SS Drgomir the bove Riemnn-Stieltjes integrl would be equivlent to Riemnn integrl nd gin vriety of bounds were obtined for f L p, b, p Inequlities of Grüss type nd number of premture vrints were exmined fully in Section 3 covering the sitution in which f exhibits t most first derivtive Applictions to numericl qudrture were investigted covering rules of Newton- Cotes type contining the evlution of the function t three possible points: the interior nd extremities The development included the midpoint, trpezoidl nd Simpson type rules However, unlike the clssicl rules, the results were not s restrictive in tht the bounds re derived in terms of the behviour of t most the first derivtive nd the Peno kernel 398) It is the im of the current section to obtin inequlities for f n) L p, b, p where f n) re gin evluted t most t n interior point x nd the end points Results tht involve the evlution only t n interior point re termed Ostrowski type nd those tht involve only the boundry points will be referred to s trpezoidl type In the numericl nlysis literture these re lso termed s Open nd Closed Newton-Cotes rules Atkinson ) respectively In 938, Ostrowski see for exmple 44, p inequlity: 468) proved the following integrl Let f : I R R be differentible mpping on I I is the interior of I), nd let, b I with < b If f :, b) R is bounded on, b), ie, f : f t) <, then we hve the inequlity: sup t,b) 399) for ll x, b f x) b f t) dt 4 + x +b b ) ) b ) f The constnt 4 is shrp in the sense tht it cnnot be replced by smller one For pplictions of Ostrowski s inequlity to some specil mens nd some numericl qudrture rules, we refer the reder to the recent pper 36 by SS Drgomir nd S Wng who used integrtion by prts from p x, t) f t) dt to prove Ostrowski s inequlity 399) where p x, t) is peno kernel given by t, t, x 300) p x, t) t b, t x, b Fink 38 used the integrl reminder from Tylor series expnsion to show tht for f n ) bsolutely continuous on, b, then the identity ) b 30) f t) dt n b ) f x) + F k x) K F x, t) f n) t) dt n is shown to hold where 30) K F x, t) x t)n n )! p x, t) n k with p x, t) being given by 300)

196 3 THREE POINT QUADRATURE RULES 88 nd F k x) n k k! x ) k f k ) ) + ) k b x) k f k ) b) Fink then proceeds to obtin vriety of bounds from 30), 30) for f n) L p, b Milovnović nd Pečrić 43 erlier obtined result for f n) L, b lthough they did not use the integrl form of the reminder It my be noticed tht 30) is gin n identity tht involves function evlutions t three points to pproximte the integrl from the resulting inequlities See Mitrinović, Pečrić nd Fink 6, Chpter XV for further relted results nd ppers 8, 3 nd 3 A number of other uthors hve obtined results in the literture tht my be recptured under the generl formultion of the current work These will be highlighted throughout the section The section is structured s follows A vriety of identities re obtined in Subsection 33 for f n ) bsolutely continuous for generlistion of the kernel 398) Specific forms re highlighted nd generlised Tylor-like expnsion is obtined Inequlities re developed in Subsection 333 nd perturbed results through Grüss inequlities nd premture vrints re discussed in Subsection 334 Subsection 335 demonstrtes the pplicbility of the inequlities to numericl integrtion Concluding remrks for the section re given in Subsection Some Integrl Identities In this subsection, identities re obtined involving n-time differentible functions with evlution t n interior point nd t the end points Theorem 359 Let f :, b R be mpping such tht f n ) is bsolutely continuous on, b Further, let α :, b, b nd β :, b, b, α x) x, β x) x Then, for ll x, b the following identity holds, 303) ) n K n x, t) f n) t) dt f t) dt n k R k x) f k ) x) + S k x), k! where the kernel K n :, b R is given by 304) K n x, t) : t αx)) n n!, t, x t βx)) n n!, t x, b, 305) { R k x) β x) x) k + ) k x α x)) k nd S k x) α x) ) k f k ) ) + ) k b β x)) k f k ) b)

197 89 P Cerone nd SS Drgomir Proof Let 306) I n x) ) n K n x, t) f n) t) dt ) n J n, x, b) then from 305) J n, x, x) x giving, upon using integrtion by prts 307) J n, x, x) Similrly, x α x))n f n ) x) n! t α x)) n f n) t) dt n! + ) n α x) ) n f n ) ) n! J n, x, x) J n x, x, b) ) n β x) x) n f n ) x) n! b β x))n + f n ) b) J n x, x, b) n! nd so upon dding to 307) gives from 306) the recurrence reltion 308) I n x) I n x) ω n x), where 309) n! ω n x) with R n x) nd S n x) being given by 305) It my esily be shown tht 30) I n x) R n x) f n ) x) + S n x) n ω k x) + I 0 x) k is solution of 308) nd so the theorem is proven since 30) is equivlent to 303) nd I n x) is s given by 306) Remrk 368 If we tke n then n identity obtined by Cerone nd Drgomir 5 results In the sme pper Riemnn-Stieltjes integrls were lso considered Remrk 369 If α x) nd β x) b then S k x) 0 nd the Ostrowski type results for n-time differentible functions of Cerone et l re recptured Merkle 4 lso obtins Ostrowski type results For α x) β x) x then R x) 0 nd the generlized trpezoidl type rules for n-time differentible functions of Cerone et l re obtined Qi 48 used Tylor series whose reminder ws not expressed in integrl form so tht only the supremum norm ws possible If the integrl form of the reminder were used, then similr to Fink 38, the other L p, b) norms for p would be possible However, this will not be pursued

198 3 THREE POINT QUADRATURE RULES 90 further here For α x) nd β x) t their respective midpoints, then the identity 3) ) n K n x, t) f n) t) dt + f t) dt n k k k! { b x) k + ) k x ) k f k ) x) x ) k f k ) ) + ) k b x) k f k ) b) results, where 3) K n x, t) t +x ) n n!, t, x t x+b ) n n!, t x, b As demonstrted in the bove remrks, different choices of α x) nd β x) give vriety of identities The following corollry llows for α x) nd β x) to be in the sme reltive position within their respective intervls Corollry 360 Let f stisfy the conditions s stted in Theorem 359 Then the following identity holds for ny γ 0, nd x, b Nmely, 33) ) n C n x, t) f n) t) dt n { f t) dt γ) k b x) k + ) k x ) k f k ) x) k! k } +γ k x ) k f k ) ) + ) k b x) k f k ) b), where 34) C n x, t) Proof Let t γx+ γ)) n n!, t, x t γx+ γ)b) n n!, t x, b 35) α x) γx + γ) nd β x) γx + γ) b, then 36) { so tht from 305) nd x α x) γ) x ), α x) γ x ) nd β x) x γ) b x), b β x) γ b x) R k x) γ) k b x) k + ) k x ) k S k x) γ k x ) k f k ) ) + ) k b x) k f k ) b) In ddition, C n x, t) is the sme s K n x, t) in 304) with α x) nd β x) s given by 35) nd hence the corollry is proven The following Tylor-like formul with integrl reminder lso holds }

199 9 P Cerone nd SS Drgomir Corollry 36 Let g :, y R be mpping such tht g n) is bsolutely continuous on, y Then for ll x, y we hve the identity 37) g y) n { g ) + β x) x) k + ) k x α x)) k g k) x) k! k } + α x) ) k g k) ) + ) k y β x)) k g k) y) y + ) n τ n x, t) g n+) t) dt where 38) τ n x, t) t αx)) n n!, t, x t βx)) n n!, t x, y Proof The proof is stright forwrd from Theorem 359 on tking f g nd b y so tht β x) x, y nd τ n x, t) K n x, t) for t, y Remrk 370 If α x) β x) x then we recpture the results of Cerone et l, trpezoidl type series expnsion Tht is, n expnsion involving the end points For α x), β x) b then Tylor-like expnsion of Cerone et l is reproduced s re the results of Merkle Integrl Inequlities In this subsection we develop some inequlities from using the identities obtined in Subsection 33 Theorem 36 Let f :, b R be mpping such tht f n ) is bsolutely continuous on, b nd, let α :, b, b nd β :, b, b, α x) x, β x) x Then the following inequlities hold for ll x, b 39) where 30) P n x) : Q n q, x) f t) dt n k R k x) f k ) x) + S k x) k! f n) n! Q n, x) if f n) L, b, f n) p n! Q n q, x) q if f n) L p, b with p >, p + q, f n) n! M n x), if f n) L, b, α x) ) nq+ + x α x)) nq+ nq + + β x) x) nq+ + b β x)) nq+,

200 3) 3 THREE POINT QUADRATURE RULES 9 M x) { b + α x) + x + β x) x + b + x + b + α x) + x + β x) x + b } mx {α x), x α x), x β x), b β x)}, R k x), S k x) re given by 305), nd f n) : ess sup f n) t) < nd t,b f n) p : ) b f n) t) p p, p Proof Tking the modulus of 303) then 3) P n x) I n x), where P n x) is s defined by the left hnd side of 39) nd 33) I n x) K n x, t) f n) t) dt, with K n x, t) given by 304) Now, observe tht 34) where, from 304), I n x) f n) K n x, ) f n) K n x, t) dt, 35) K n x, t) dt n! + { αx) βx) x n + )! x t α x) n dt + t α x) n dt αx) } t β x) n dt + t β x) n dt βx) α x) ) n+ + x α x)) n+ + β x) x) n+ + b β x)) n+ Thus, on combining 3), 34) nd 35), the first inequlity in 39) is obtined Further, using Hölder s integrl inequlity we hve the result 36) I n x) f n) p K n x, ) q where p +, with p > q f n) p K n x, t) q dt ) q

201 93 P Cerone nd SS Drgomir Now, 37) K n x, t) q dt n! + { αx) x t α x) nq dt + t α x) nq dt αx) } t β x) nq dt + t β x) nq dt x βx) βx) n! Q n q, x), where Q n q, x) is s given by 30) Combing 37) with 36) gives the second inequlity in 39) Finlly, let us observe tht from 3) 38) where I n x) f n) K n x, ) f n) sup t,b K n x, t) f n) mx { α x) n, x α x) n, b β x) n, x β x) n } n! f n) M n x), n! 39) M x) mx {M x), M x)} with nd The well known identity M x) mx {α x), x α x)} M x) mx {β x) x, b β x)} 330) mx {X, Y } X + Y my be used to give 33) + X Y M x) x + α x) +x, nd M x) b x + β x) x+b Using the identity 330) gin gives, from 39), M x) M x) + M x) + M x) M x) which on substituting 33) gives 3) nd so from 38) nd 3) redily results in the third inequlity in 39) nd the theorem is completely proved Remrk 37 Vrious choices of α ) nd β ) llow us to reproduce mny of the erlier inequlities involving function nd derivtive evlutions t n interior point nd/or boundry points For other relted results see Chpter XV of 44

202 3 THREE POINT QUADRATURE RULES 94 If α x) nd β x) b then S k x) 0 nd Ostrowski type results for n- time differentible functions of Cerone et l re reproduced see lso 49) Further, tking n recptures the results of Drgomir nd Wng nd n gives the results of Cerone, Drgomir nd Roumeliotis 8-0 The n cse is of importnce since with x +b the clssic midpoint rule is obtined However, here the bound is obtined for f L p, b for p rther thn the trditionl f L, b, see for exmple 4 nd 6 If α x) β x) x then R x) 0 nd inequlities re obtined for generlised trpezoidl type rule in which functions re ssumed to be n-time differentible, recpturing the results in Cerone et l Tking n the clssic trpezoidl rule in which the bound involves the behviour of the second derivtive is recptured s presented in Drgomir et l 7 Tking α ) nd β ) to be other thn t the extremities results in three point inequlities for n time differentible functions Cerone nd Drgomir 5 presented results for functions tht t most dmit first derivtive Remrk 37 It should be noted tht the bounds in 39) my themselves be bounded since α ), β ) nd x hve not been explicitly specified To demonstrte, consider the mppings, for t A, B, h t) t A) θ + B t) θ, θ > 33) nd h t) B A + t A+B Now, both these functions ttin their mximum vlues t the ends of the intervl nd their minimums t the midpoints Tht is, they re symmetric nd convex Thus, sup h t) h A) h B) B A) θ, t A,B 333) sup h t) h A) h B) B A, t A,B inf h t) h A+B ) t A,B B A ) θ, nd inf h t) h A+B ) t A,B B A Using 33) nd 333) then from 30) nd 3), on tking α ) nd β ) t either of their extremities gives Q n q, x) Q U n q, x) x ) nq+ + b x) nq+ b )nq+ nq + nq + nd M x) M U x) b + x b, where the corsest bounds re obtined from tking x t its extremities The following corollry holds

203 95 P Cerone nd SS Drgomir Corollry 363 Let the conditions of Theorem 36 hold Then the following result is vlid for ny x, b Nmely, n k { 334) f t) dt b x) k + ) k x ) k f k ) x) k! k + x ) k f k ) ) + ) k b x) k f b) } k ) f n) n! n x ) n+ + b x) n+, f n) L, b, f n) p n! n nq+) q x ) nq+ + b x) nq+ q f n) L p, b with p >, p + q, f n) n! n b + x +b n, f n) L, b Proof Tking α ) nd β ) t their respective midpoints, nmely α x) +x nd β x) x+b in 39)-3) nd using 305) redily gives 334) Remrk 373 Corollry 363 could hve equivlently been proven using 3) nd 3) following essentilly the sme proof of Theorem 36 from using identity 3) The more generl setting however, llows greter flexibility nd, it is rgued, is no more difficult to prove Corollry 364 Let the conditions on f of Theorem 36 hold Then the following result for ny x, b, is vlid Nmely, for ny γ 0, n ) k f t) dt γ) k r k x) f k ) x) + γ k 335) s k x) k! where k f n) n+)! H γ) G x), f n) L, b, f n) p n!nq+) q H q q γ) G q q x), f n) L p, b p >, p + q, f n) n! ν n x), f n) L, b, H q γ) γ nq+ + γ) nq+, 336) G q x) x ) nq+ + b x) nq+, ν x) + γ b + x +b, r k x) b x) k + ) k x ) k, nd s k x) x ) k f k ) ) + ) k b x) k f k ) b)

204 3 THREE POINT QUADRATURE RULES 96 Proof Tke α ) nd β ) to be convex combintion of their respective boundry points s given by 35) then from 39)-3) nd using 36) nd 305) redily produces the stted result We omit ny further detils Remrk 374 It is instructive to note tht the reltive loction of α ) nd β ) is the sme in Corollry 364 nd is determined through the prmeter γ s defined in 35) Theorem 36 is much more generl From 35) it my be seen tht α x) β x) x is equivlent to γ, giving trpezoidl type rules while α x), β x) b corresponds to γ 0 which produces interior point rules Tking γ 0 nd γ reproduces the results of Cerone et l nd respectively Tking γ in 335) produces the optiml rule while keeping x generl nd thus reproducing the result of Corollry 363 Following the discussion in Remrk 37 nd s my be scertined from 336) the optiml rules, in the sense of providing the tightest bounds, re obtined by tking γ nd x t their respective midpoints The following two corollries my thus be stted Corollry 365 Let the conditions on f of Theorem 36 be vlid Then for ny γ 0,, the following inequlities hold 337) f t) dt n ) k ) ) ) γ) k + b + b + b r k f k ) + γ k s k k! k 338) where f n) n+)! f n) p H q n!nq+) q f n) n! ν n +b H γ) G +b ), f n) L, b, q γ) G q +b ), f n) L p, b H q γ) is s given by 336), p >, p + q, ), f n) L, b, 339) G q +b ν +b r k +b ) b ) nq+, ) b nd s k +b ) + γ ), ) b ) k + ) k ) b ) k f k ) ) + ) k f k ) b) Proof The proof is trivil Tking x +b in 335)-336) redily produces the result

205 97 P Cerone nd SS Drgomir Remrk 375 It is of interest to note from 339) tht ) + b 340) r k ) b k, k odd 0, k even so tht only the evlution of even order derivtives re involved in 337) Further, for f k ) ) f k ) b) then ) ) ) + b + b + b 34) s k f k ) ) r k f k ) b) r k so tht only evlution of even order derivtives t the end points re present Corollry 366 Let the conditions on f of Theorem 36 hold Then the following inequlities re vlid 34) f t) dt f n) n+)! f n) p n!nq+) q f n) n! n ) k ) ) ) + b + b + b k r k f k ) + s k k! k ) n ) b n+, f n) L, b, n ) q b b nq+ ) n, f n) L p, b with p >, p + q, b ) n 4, f n) L, b, where r k +b ) nd +b ) sk re s given by 339) Proof Tking γ in Corollry 365 will produce inequlities with the tightest bounds s given in 34) Alterntively, tking γ +b nd x in Corollry 364 will produce the results 34) The results 340) nd 34) together with the discussion in Remrk 375 re lso vlid for Corollry 366 The following re Tylor-like inequlities which re of interest see 4 nd 6 for relted results)

206 3 THREE POINT QUADRATURE RULES 98 Corollry 367 Let g :, y R be mpping such tht g n) is bsolutely continuous on, y Then for ll x, y 343) n g y) g ) { β x) x) k + ) k x α x)) k g k) x) k! k + α x) ) k g k) ) + ) k y β x)) k g y) } k) g n+) n! g n+) p n! g n+) n! Q n, x), g n+) L, b, Qn q, x) q, g n+) L p, b with p >, p + q, M n x), g n+) L, b, where Q n q, x) α x) ) nq+ + x α x)) nq+ nq + + β x) ) nq+ + y β x)) nq+, M x) { y + α x) + x + β x) x + y + x + y + α x) + x + β x) x + y } Proof The proof follows from Theorem 36 on tking f ) g ) nd b y so tht β x) x, y Alterntively, strting from 37) nd 38) nd, following the proof of Theorem 36 with b replced by y nd f ) replced by g ) redily produces the results shown nd the corollry is thus proven Remrk 376 Similr corollries to 363, 364, 374 nd 375 could be determined from the Tylor-like inequlities given in Corollry 366 This would simply be done by tking specific forms of α ), β ) or vlues of x s pproprite Remrk 377 If in prticulr we tke α x) nd β x) y in 343) then for ny x, y 344) n g y) g ) y x) k + ) k x ) k g k) x) k! e n x, y) k

207 99 P Cerone nd SS Drgomir 345) : g n+) n+)! g n+) p n!nq+) q g n+) n! x ) n+ + y x) n+, g n+) L, y, x ) nq+ + y x) nq+ q, g n+) L p, y with p >, p + q, + x +y n, g n+) L, y y Merkle 4 effectively obtins the first bound in 344) It is well known see for exmple, Drgomir 6) tht the clssicl Tylor expnsion round point stisfies the inequlity n g y) y ) k g k) 346) ) k! k y y u) n g n+) u) du n! : E n y), where 347) E n y) for y nd y I R y ) n+ n+)! y ) n+ q n!nq+) q y ) n n! g n+), g n+) L, y, g n+) p, g n+) L p, y with p >, p + q, g n+), g n+) L, y, Now, it my redily be noticed tht if x in 344), then the clssicl result s given by 346) is regined As discussed in Remrk 37 the bounds re convex so tht corse bound is obtined t the end points nd the best t the midpoint Thus, tking x +y gives 348) g y) g ) + y e n ), y n + ) k k k! ) k y ) k + y g k) g n+) n+)! n y ) n+, g n+) L, y, g n+) p n!nq+) q n y ) n+ q, g n+) L p, y with p >, p + q, g n+) n! n y ) n, g n+) L, y

208 3 THREE POINT QUADRATURE RULES 00 The bove inequlities 348) show tht for g C, b the series + ) k ) g ) + k! k y ) k + y g k) k converges more rpidly to g y) thn the usul one y ) k g k) ), k! k0 which comes from Tylor s expnsion 346) It should further be noted tht 347) only involves the odd derivtives of g ) evluted t the midpoint of the intervl under considertion Remrk 378 If α x) β x) x in 343), then for ny x, y n g y) g ) x ) k g k) ) + ) k y x) k g k) 349) y) k! e n x, y), k where e n x, y) is s defined by 344) See Cerone et l for relted results 334 Perturbed Rules Through Grüss Type Inequlities In 935, G Grüss see for exmple 44), proved the following integrl inequlity which gives n pproximtion for the integrl of product in terms of the product of integrls Theorem 368 Let h, g :, b R be two integrble mppings so tht φ h x) Φ x) nd γ g x) Γ for ll x, b, where φ, Φ, γ, Γ re rel numbers Then we hve 350) T h, g) Φ φ) Γ γ), 4 where 35) T h, g) b h x) g x) dx b h x) dx b g x) dx nd the inequlity is shrp, in the sense tht the constnt 4 cnnot be replced by smller one For simple proof of this fct s well s for extensions, generlistions, discrete vrints nd other ssocited mteril, see 44, nd the ppers 7, 3, 5, nd 35 where further references re given A premture Grüss inequlity is embodied in the following theorem which ws proved in the pper 4 It provides shrper bound thn the bove Grüss inequlity The term premture is used to denote the fct tht the result is obtined from not completing the proof of the Grüss inequlity if one of the functions is known explicitly See lso 5 for further detils Theorem 369 Let h, g be integrble functions defined on, b nd let d g t) D Then 35) T h, g) D d T h, h),

209 0 P Cerone nd SS Drgomir where T h, g) is s defined in 35) The bove Theorem 369 will now be used to provide perturbed generlised three point rule 335 Perturbed Rules From Premture Inequlities We strt with the following result Theorem 370 Let f :, b R be such tht the derivtive f n ), n is bsolutely continuous on, b Assume tht there exist constnts γ, Γ R such tht γ f n) t) Γ e on, b Then the following inequlity holds n 353) ρ n x) : f t) dt R k x) f k ) x) + S k x) k! k ) n θ n x) n + f n ) b) f n ) ) b Γ γ I x, n) n! Γ γ n b )n+, n + )! n + where 354) I x, n) { n + ) n b ) ˆQ n, x) n + 4 } + n + ) z i z j zi n z j ) n i j>i Z {α x), x α x), β x) x, b β x)}, z i Z, i,, 4, ˆQ n, x) n + ) Q n, x) with Q n, x) being s defined in 30), θ n x) ) n z n+ + z n+ + ) n z n+ 3 + z n+ 4, nd R k x), S k x) re s given by 305) Proof Applying the premture Grüss result 35) by ssociting f n) t) with g t) nd h t) with K n x, t), from 304) gives 355) )n K n x, t) f n) t) dt ) b ) n f n ) b) f n ) ) K n x, t) dt b b ) Γ γ T K n, K n ), where from 35) T K n, K n ) b K n x, t) dt b b K n x, t) dt)

210 3 THREE POINT QUADRATURE RULES 0 Now, from 304), 356) b b K n x, t) dt x t α x)) n n! dt + x t β x)) n dt n! x α x)) n+ + ) n α x) ) n+ b ) n + )! b β x)) n+ + ) n β x) x) n+ nd 357) : b b ) n + )! θ n x) b ) n!) Kn x, t) dt x t α x)) n dt + b ) n!) n + ) b β x)) n+ + β x) x) n+ x t β x)) n dt x α x)) n+ + α x) ) n+ on using 30) ˆQ b ) n!) n, x) n + ) Hence, substitution of 356) nd 357) into 355) gives K n x, t) f n) t) dt ) n θ n x) n + )! f n ) b) f n ) ) 358) b where 359) Now, let Γ γ J x, n), n! n + ) n + ) J x, n) n + ) b ) ˆQ n, x) n + ) θ n x) 360) A α x), X x α x), Y β x) x nd B b β x), then 356) nd 357) imply tht ˆQ n, x) A n+ + X n+ + Y n+ + B n+ nd θ n x) ) n A n+ + X n+ + ) n Y n+ + B n+

211 03 P Cerone nd SS Drgomir Hence, from 359) nd using the fct tht b A + X + Y + B, 36) n + ) b ) ˆQ n, x) n + ) θ n x) n ˆQn, x) + n + ) A + X + Y + B) Q n, x) θ n x) n ˆQn, x) + n + ) 4 z i z j zi n z j ) n i j>i fter some stright forwrd lgebr, where Z {A, X, Y, B}, z i Z, i,, 4 Substitution of 36) into 359) gives I x, n) Jx,n) n+) n+ s presented by 354) Utilising identity 303) in 355) gives 353) nd the first prt of the theorem is proved The upper bound is obtined by tking α ), β ), x t their end points since I x, n) is convex nd symmetric The second term for I x, n) is then zero nd ˆQ n, x) < b ) n+ nd hence fter some simplifiction, the theorem is completely proven Corollry 37 Let the conditions of Theorem 368 hold Then the following result is vlid, n k 36) f t) dt r k x) f k ) x) + s k x) k! k n + ) n A n + B n ) f n ) b) f n ) ) b Γ γ { n+) 4n + + ) n ) n + ) A n+) + B n+) n! + 4n + n + ) AB A n + B n) + 4 n + ) ) n A B) n+ }, where r m x) nd s m x) re s given by 336) nd A x, B b x Proof Let α x) +x nd β x) x+b in 353), redily giving the left hnd side of 36) Now, for the right hnd side Tking A x, B b x, we hve nd ˆQ n, x) n A n+ + B n+ 4 z i z j zi n z j ) n n+) A n+) + B n+) + ) n ) i j>i +AB A n + ) n B n)

212 3 THREE POINT QUADRATURE RULES 04 so tht from 354) nd using the fct tht b A + B, 363) n + ) n + I x, n) { n+) 4n A + B) A n+ + B n+ + n + ) A n+) + B n+)) + ) n ) +AB A n + ) n B n) } { n+) 4n + + ) n ) n + ) A n+) + B n+) + 4n + n + ) AB A n + B n) + 4 n + ) ) n AB) n+} A simple substitution in 353) of 363) completes the proof Corollry 37 Let the conditions of Theorem 368 nd Corollry 37 hold Then the following inequlity results, n k ) ) ) + b + b + b 364) f t) dt r k f k ) + s k k! k 4 n + ) n ) f n ) b) f ) n ) where r m +b b Γ γ b n! 4 ) nd sm +b ) n+) 8n + 3 n + ) + ) n ), ) re s given in 339) Proof The proof follows directly from 36) with x +b so tht A B, giving for the brces on the right hnd side ) n+) b 8n + 3 n + ) + ) n ) 4 Some stright forwrd simplifiction produces the result 364) Remrk 379 It my be noticed See lso Remrk 375) tht only even order degrees re involved, in 364), t the midpoint while this is only the cse t the endpoints if the further restriction f k ) ) f k ) b) is imposed Further, if n is odd, then there is no perturbtion rising from the Grüss type result 364) Theorem 373 Let the conditions of Theorem 368 be stisfied Further, suppose tht f n) is bsolutely continuous nd is such tht f n+) : ess sup f n+) t) < t,b Then 365) ρ n x) b f n+) I x, n), n! where ρ n x) is the perturbed interior point rule given by the left hnd side of 353) nd I x, n) is s given by 354)

213 05 P Cerone nd SS Drgomir Proof Let h, g :, b R be bsolutely continuous nd h, g be bounded Then Chebychev s inequlity holds see 47, p 07) T h, g) b ) sup h t) sup g t) t,b t,b Mtić, Pečrić nd Ujević 4 using premture Grüss type rgument proved tht 366) T h, g) b ) sup g t) T h, h) t,b Thus, ssociting f n) ) with g ) nd K x, ), from 304), with h ) in 366) produces 365) where I x, n) is s given by 354) Theorem 374 Let the conditions of Theorem 368 be stisfied Further, suppose tht f n) is loclly bsolutely continuous on, b) nd let f n+) L, b) Then 367) ρ n x) b π f n+) I x, n), n! where ρ n x) is the perturbed generlised interior point rule given by the left hnd side of 353) nd I x, n) is s given in 354) Proof The following result ws obtined by Lupş see 47, p 0) For h, g :, b) R loclly bsolutely continuous on, b) nd h, g L, b), then where T h, g) k : b b ) π h g, k t) ) for k L, b) Mtić, Pečrić nd Ujević 4 further show tht 368) T h, g) b π g T h, h) Now, ssociting f n) ) with g ) nd K x, ), from 304) with h ) in 368) gives 367) where I x, n) is s found in 354) Remrk 380 Results 365) nd 367) re not redily comprble to tht obtined in Theorem 368 since the bound now involves the behviour of f n+) ) rther thn f n) ) Remrk 38 Premture results presented in this subsection my lso be obtined, producing bounds for generlized Tylor-like series expnsion by tking f g nd b y See lso Mtić et l 4 for relted results

214 3 THREE POINT QUADRATURE RULES Applictions in Numericl Integrtion Any of the inequlities in Subsections 333 nd 334 my be utilised for numericl implementtion Here we illustrte the procedure by giving detils for the implementtion of Corollry 364 Consider the prtition I m : x 0 < x < < x m < x m b of the intervl, b nd let the intermedite points ξ ξ 0,, ξ m ) where ξj x j, x j+ for j 0,,, m Define the formul for γ 0,, 369) A m,n f, I m, ξ) m j0 k n ) k { γ) k ) r k ξj f k ) ) ξ j k! +γ k A k j f k ) x j ) + ) k B k j f k ) x j+ ) }, where 370) r k ξj ) B k j + ) k A k j A j ξ j x j, B j x j+ ξ j, nd h j A j + B j x j+ x j for j 0,,, m The following theorem holds involving 369) Theorem 375 Let f :, b R be mpping such tht f n ) is bsolutely continuous on, b nd I m be prtition of, b s described bove Then the following qudrture rule holds Nmely, 37) f x) dx A m,n f, I m, ξ) + R m,n f, I m, ξ), where A m,n is s defined by 369)-370) nd the reminder R m,n f, I m, ξ) stisfies the estimtion 37) R m,n f, I m, ξ) f n) n+)! f n) p n! f n) n! H γ) m H qγ) nq+) q j0 m j0 A n+ j + B n+ ) j, for f n) L, b, A nq+ j + γ ) n νh) + mx j0,,m ) q + B nq+ j, for f n) L p, b, p >, p + q, n ξ j xj+xj+, for f n) L, b, where H q γ) is given by 336), ν h) mx {h j j 0,, m }, of the terms re s given in 370) nd the rest

215 07 P Cerone nd SS Drgomir Proof Apply Corollry 364 on the intervl x j, x j+ to give xj+ n ) k { f t) dt γ) k ) r k ξj f k ) ) 373) ξ k! j x j k +γ k A k j f k ) x j ) + ) k B k j f k ) x j+ ) } H γ) n+)! H qγ) n! sup f n) t) A n+ j + B n+ ) j, t x j,x j+ xj+ x j f n) u) p p du A nq+ j +B nq+ j nq+ ) q, + γ ) n xj+ n! x j f n) u) hj du + ξj xj+xj+ n ), where the prmeters re s defined in 370) nd H q γ) is s given in 336) Summing over j from 0 to m nd using the generlised tringle inequlity gives 374) R m,n f, I m, ξ) m H γ) n+)! j0 sup t x j,x j+ f n) t) A n+ j + B n+ ) j, Now, since follows H qγ) n! m j0 + γ ) n n! xj+ x j m j0 f n) u) ) p p du x j A nq+ j +B nq+ j nq+ ) q, xj+ f n) u) ) hj du + ξj n xj+xj+ ) sup f n) t) f n), the first inequlity in 37) redily t x j,x j+ Further, using the discrete Hölder inequlity, we hve m j0 xj+ f n) u) p du x j ) q nq + ) p A nq+ j + B nq+ j nq + m xj+ f n) u) p du j0 m A nq+ j0 f n) p nq + ) q j m j0 x j + B nq+ j A nq+ j ) q q + B nq+ j q ) q ) p ) q p p

216 3 THREE POINT QUADRATURE RULES 08 nd thus the second inequlity in 37) is proven Finlly, let us observe from 373) tht m j0 ) xj+ hj f n) u) du + ξ j x j + x j+ x j + ξ j x ) j + x j+ n m j0 x j ν h) + mx j0,,m ξ j x ) j + x j+ n f n) hj mx j0,,m xj+ ) n f n) u) du Hence, the theorem is completely proved Remrk 38 Following the discussion in Remrk 37, corser upper bounds to those in 37) re obtined by tking ξ j t either extremity of its intervl, giving f n) n + )! H γ) f n) ν n h) n! m j0 h n+ j, f n) p n! H q γ) nq + ) q m j0 h nq+ j q, for f n) belonging to the obvious L p, b, p These re uniform bounds reltive to the intermedite points ξ Corollry 376 Let the conditions of Theorem 375 hold Then we hve f x) dx A m,n f, I m ) + R m,n f, I m ), where A m,n f, I m ) m j0 k +γ k h k j n ) k { γ) k r k δ j ) f k ) δ j ) k! f k ) x j ) + ) k f k ) x j+ ) }, with δ j x j + x j+ nd r k δ j ) hk j + ) k )

217 09 P Cerone nd SS Drgomir nd the reminder R m,n f, I m ) stisfies the inequlity R m,n f, I m ) f n) n+)! f n) p n! f n) n! m j0 H qγ) nq+) q ) k hj, for f n) L, b, m j0 + γ ) q nq+ hj, for f n) L p, b ) n νh) p >, p + q ), n, for f n) L, b Proof The proof is trivil from Theorem 375 Tking ξ j xj+xj+ gives A j B j hj nd the results stted follow 337 Concluding Remrks Tking γ 0 in Corollry 360 gives generlised Ostrowski type identity which hs bounds given by Corollry 364 with γ 0, reproducing the results of Cerone nd Drgomir 3 This gives corse upper bound s discussed in Remrk 37 since the bound is convex nd symmetric in both γ nd x Let the identity be denoted by M n x) which is produced from tking Peno kernel of 375) k M x, t) t ) n n!, t, x t b) n n!, t x, b Further, tking γ in Corollry 360 gives generlised Trpezoidl type identity with bounds given by Corollry 364 with γ reproducing the results of Cerone nd Drgomir 6 This choice of γ gin gives the corsest bound s discussed in Remrk 37 Let the resulting identity be denoted by T n x), which results from tking Peno kernel of 376) k T x, t) x t)n n! It ws shown in Cerone nd Drgomir 3 tht k M x, t) q k T x, t) q Let which is obtined from the kernel I L x) λm n x) + λ) T n x) k x, t) λk M x, t) + λ) k T x, t) where k M x, t) nd k T x, t) re given by 375) nd 376) respectively The best one cn do for q >, q with such kernel when determining bounds is to use the tringle inequlity nd so 377) k x, t) q λ k M x, t) q + λ) k T x, t) q k M x, t) q k T x, t) q

218 3 THREE POINT QUADRATURE RULES 0 The results thus obtined would be given by, for f n) L p, b, p, n n f t) dt λ ) k r k x) f k ) x) λ) ) k s k x) k f n) n+)! G x), for f n) L, b, f n) p n!nq+) q f n) n! G q x), for f n) L p, b, p >, p + q, + x +b, for f n) L, b, b where r k x), s k x), G q x) re given by 336) nd it should be noted tht the bound is independent of λ This result would be no more difficult to implement thn 335) in Corollry 364, with the best bounds resulting from γ For q, or infinity, k x, t) q my be evluted explicitly without using the tringle inequlity t which stge comprison with the results of Corollry 364 would be less conclusive This will not be discussed further here In the ppliction of the current work to qudrture, if we wished to pproximte the integrl f x) dx using rule Q f, I m) with bound E m), where I m is uniform prtition for exmple, with n ccurcy of ε > 0, then we require m ε N where m ε E ε) +, with w denoting the integer prt of w R The pproch thus described enbles the user to predetermine the prtition required to ssure the result to be within certin error tolernce This pproch is somewht different from tht commonly used of systemtic mesh refinement followed by comprison of successive pproximtions which forms the bsis of stopping rule See, 37 nd 39 for comprehensive tretment of trditionl methods Acknowledgement The uthors would like to thnk John Roumeliotis for his comments nd his help in the drwing of Figure 3 nd Josip Pečrić for n ide tht led to the premture Grüss result of Subsection 33 k

219 Bibliogrphy KE ATKINSON, An Introduction to Numericl Anlysis, Wiley nd Sons, Second Edition, 989 N S BARNETT nd SS DRAGOMIR, An inequlity of Ostrowski type for cumultive distribution functions, Kyungpook Mth J, 39) 999), P CERONE nd S S DRAGOMIR, Midpoint type rules from n inequlities point of view, Hndbook of Anlytic-Computtionl Methods in Applied Mthemtics, GA Anstssiou Ed), CRC Press, New York 000), P CERONE nd S S DRAGOMIR, Three point identities nd inequlities for n time differentible functions, SUT Journl of Mthemtics, in press) 5 P CERONE nd S S DRAGOMIR, Three point qudrture rules involving, t most, first derivtive, submitted, RGMIA Res Rep Coll, 4, 999), Article 8 ON- LINEhttp://rgmivueduu/vn4html 6 P CERONE nd S S DRAGOMIR, Trpezoidl type rules from n inequlities point of view, Hndbook of Anlytic-Computtionl Methods in Applied Mthemtics, GA Anstssiou Ed), CRC Press, New York 000), P CERONE, S S DRAGOMIR nd J ROUMELIOTIS, An inequlity of Ostrowski-Grüss type for twice differentible mppings nd pplictions, Preprint RGMIA Res Rep Coll, ) 998), Article 8 ONLINEhttp://rgmivueduu/vnhtml 8 P CERONE, S S DRAGOMIR nd J ROUMELIOTIS, An inequlity of Ostrowski type for mppings whose second derivtives re bounded nd pplictions, Est Asin J of Mth, 5) 999), -9 Preprint RGMIA Res Rep Coll, ) 998), Article 4 ON- LINEhttp://rgmivueduu/vnhtml 9 P CERONE, S S DRAGOMIR nd J ROUMELIOTIS, An Ostrowski type inequlity for mppings whose second derivtives belong to L p, b) nd pplictions, Preprint RGMIA Res Rep Coll, ) 998), Article 5 ONLINE 0 P CERONE, S S DRAGOMIR nd J ROUMELIOTIS, On Ostrowski type for mppings whose second derivtives belong to L, b) nd pplictions, Honm Mth J, ) 999), -3 Preprint RGMIA Res Rep Coll, ) 998), Article 7 ONLINE P CERONE, S S DRAGOMIR nd J ROUMELIOTIS, Some Ostrowski type inequlities for n-time differentible mppings nd pplictions, Demonstrtio Mth, 3) 999), Preprint RGMIA Res Rep Coll, ) 998), 5-66 ONLINE P CERONE, SS DRAGOMIR, J ROUMELIOTIS nd J SUNDE, A new generliztion of the trpezoid formul for n-time differentible mppings nd pplictions, Demonstrtio Mth, in press RGMIA Res Rep Coll, 5) 999), Article 7 ONLINE 3 SS DRAGOMIR, A Grüss type integrl inequlity for mppings of r-hölder s type nd pplictions for trpezoid formul, Tmkng J of Mth, 3) 000), SS DRAGOMIR, A Tylor like formul nd ppliction in numericl integrtion, submitted) 5 SS DRAGOMIR, Grüss inequlity in inner product spces, The Austrlin Mth Gzette, 6) 999), ONLINE 6 SS DRAGOMIR, New estimtion of the reminder in Tylor s formul using Grüss type inequlities nd pplictions, Mth Ineq nd Appl, ) 999), SS DRAGOMIR, On Simpson s Qudrture Formul for Lipschitzin Mppings nd Applictions, Soochow J of Mth, 5) 999), 75-80

220 BIBLIOGRAPHY 8 SS DRAGOMIR, On Simpson s Qudrture Formul for Mppings with Bounded Vrition nd Applictions, Tmkng J of Mth, ), SS DRAGOMIR, On the Ostrowski s Integrl Inequlity for Mppings with Bounded Vrition nd Appliction, Mth Ineq & Appl, in press) 0 SS DRAGOMIR, On the trpezoidl qudrture formul for mppings with bounded vrition, Krgujevc J Mth, 000), 3-9 SS DRAGOMIR, Ostrowski s Inequlity for Monotonous Mppings nd Applictions, J KSIAM, 3) 999), 7-35 SS DRAGOMIR, Some integrl inequlities of Grüss type, Indin J of Pure nd Appl Mth, 34) 000), ONLINE 3 SS DRAGOMIR, The Ostrowski s Integrl Inequlity for Lipschitzin Mppings nd Applictions, Computers nd Mth with Applic, ), SS DRAGOMIR nd N S BARNETT, An Ostrowski type inequlity for mppings whose second derivtives re bounded nd pplictions, J Indin Mth Soc, ), Preprint RGMIA Res Rep Coll, ) 998), Article 9 ON- LINEhttp://rgmivueduu/vnhtml 5 SS DRAGOMIR, P CERONE nd CEM PEARCE, Generliztion of the Trpezoid Inequlity for Mppings with Bounded Vrition nd Applictions, Tmkng J Mth, in press) 6 SS DRAGOMIR, P CERONE nd A SOFO, Some remrks on the midpoint rule in numericl integrtion, Studi Mth Bbes-Bolyi Univ, in press) ONLINE 7 SS DRAGOMIR, P CERONE nd A SOFO, Some remrks on the trpezoid rule in numericl integrtion, Indin J of Pure nd Appl Mth, 35) 000), Preprint: RGMIA Res Rep Coll, 5) 999), Article ON- LINEhttp://rgmivueduu/vn5html 8 SS DRAGOMIR, YJ CHO nd SS KIM, Some remrks on the Milovnović-Pečrić Inequlity nd in Applictions for specil mens nd numericl integrtion, Tmkng J of Mth, 30 3) 999), 03-9 SS DRAGOMIR nd I FEDOTOV, An Inequlity of Grüss type for Riemnn-Stieltjes Integrl nd Applictions for Specil Mens, Tmkng J of Mth, 4, 9 998), SS DRAGOMIR nd A McANDREW, On Trpezoid Inequlity vi Grüss Type Result nd Applictions, Tmkng J of Mth, in press) 3 SS DRAGOMIR, JE PEČARIĆ nd S WANG, The unified tretment of trpezoid, Simpson nd Ostrowski type inequlity for monotonic mppings nd pplictions, Mth nd Comp Modelling, 3 000), 6-70 Preprint: RGMIA Res Rep Coll, 4) 999), Article 3 ONLINEhttp://rgmivueduu/vn4html 3 SS DRAGOMIR nd A SOFO, An integrl inequlity for twice differentible mppings nd pplictions, Preprint: RGMIA Res Rep Coll, ), Article 9, 999 ON- LINEhttp://rgmivueduu/vnhtml 33 SS DRAGOMIR nd S WANG, A new inequlity of Ostrowski s type in L norm nd pplictions to some specil mens nd some numericl qudrture rules, Tmkng J of Mth, 8 997), SS DRAGOMIR nd S WANG, A new inequlity of Ostrowski s type in L p norm, Indin J of Mth, 403) 998), SS DRAGOMIR nd S WANG, An inequlity of Ostrowski-Grüss type nd its pplictions to the estimtion of error bounds for some specil mens nd for some numericl qudrture rules, Computers Mth Applic, ), 5-36 SS DRAGOMIR nd S WANG, Applictions of Ostrowski s inequlity to the estimtion of error bounds for some specil mens nd to some numericl qudrture rules, Appl Mth Lett, 998), H ENGELS, Numericl Qudrture nd Cubture, Acdemic Press, New York, A M FINK, Bounds on the derivtion of function from its verges, Czech Mth Journl, 4 99), No 7, AR KROMMER nd CW UEBERHUBER, Numericl Integrtion on Advnced Computer Systems, Lecture Notes in Computer Science, 848, Springer-Verlg, Berlin, A LUPAŞ, The best constnt in n integrl inequlity, Mthemtic, 538) ) 973), 9-

221 3 P Cerone nd SS Drgomir 4 M MATIĆ, JE PEČARIĆ nd N UJEVIĆ, On new estimtion of the reminder in generlised Tylor s formul, Mth Ineq Appl, 3) 999), M MERKLE, Remrks on Ostrowski s nd Hdmrd s inequlity, Univ Beogrd Publ Elecktrotechn Ser Mt, 0 999), GV MILOVANOVIĆ nd JE PEČARIĆ, On generlistions of the inequlity of A Ostrowski nd some relted pplictions, Univ Beogrd Publ Elecktrotechn Ser Fiz, DS MITRINOVIĆ, JE PEČARIĆ nd AM FINK, Clssicl nd New Inequlities in Anlysis, Kluwer Acdemic Publishers, DS MITRINOVIĆ, JE PEČARIĆ nd AM FINK, Inequlities for Functions nd Their Integrls nd Derivtives, Kluwer Acdemic Publishers, A OSTROWSKI, Uber die Absolutbweichung einer differentienbren Funktionen von ihren Integrlmittelwert, Comment Mth Hel, 0 938), JE PEČARIĆ, F PROSCHAN nd YL TONG, Convex Functions, Prtil Orderings, nd Sttisticl Applictions, Acdemic Press, F QI, Further generlistions of inequlities for n integrl, Univ Beogrd Publ Elecktrotechn Ser Mt, 8 997), J SÁNDOR, On the Jensen-Hdmrd inequlity, Studi Univ Bbes-Bolyi, Mth, 36) 99), 9-5

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223 CHAPTER 4 Product Brnches of Peno Kernels nd Numericl Integrtion by P CERONE Abstrct Product brnches of Peno kernels re used to obtin results suitble for numericl integrtion In prticulr, identities nd inequlities re obtined involving evlutions t n interior nd t the end points It is shown how previous work nd rules in numericl integrtion re recptured s prticulr instnces of the current development Explicit priori bounds re provided llowing the determintion of the prtition required for chieving prescribed error tolernce In the min, Ostrowski-Grüss type inequlities re used to obtin bounds on the rules in terms of vriety of norms 4 Introduction Recently, Cerone nd Drgomir 7 obtined the following identity involving n time differentible functions with evlution t n interior point nd t the end points For f :, b R mpping such tht f n ) is bsolutely continuous on, b with α :, b R, β :, b R, α x) x nd β x) x, then for ll x, b the following identity holds 4) ) n f t) dt K n x, t) f n) t) dt n k R k x) f k ) x) + S k x), k! where the kernel Kn :, b R is given by 4) Kn x, t) : t αx)) n n!, t, x t βx)) n n!, t x, b, 5

224 4 PRODUCT BRANCHES OF PEANO KERNELS AND NUMERICAL INTEGRATION 6 43) nd R k x) β x) x) k + ) k x α x)) k S k x) α x) ) k f k ) ) + ) k b β x)) k f k ) b) They obtined inequlities for f n) L p, b, p In n erlier pper 6 the sme uthors treted the cse n but lso exmined the results eminting from the Riemnn-Stieltjes integrl K x, t) df t) nd obtined bounds for f being of bounded vrition, Lipschitzin or monotonic Applictions to numericl qudrture were investigted covering rules of Newton-Cotes type contining the evlution of the function t three possible points: the interior nd extremities The development included the midpoint, trpezoidl nd Simpson type rules However, unlike the clssicl rules, the results were not s restrictive in tht the bounds were derived in terms of the behviour of t most the first derivtive nd the Peno kernel K x, t) Perturbed rules were lso obtined using Grüss type inequlities In 938, Ostrowski see for exmple 6, p inequlity: 468) proved the following integrl Let f : I R R be differentible mpping on I I is the interior of I), nd let, b I with < b If f :, b) R is bounded on, b), ie, f : f t) <, then we hve the inequlity: sup t,b) 44) for ll x, b f x) b f t) dt 4 + x +b b ) ) b ) f The constnt 4 is shrp in the sense tht it cnnot be replced by smller one For pplictions of Ostrowski s inequlity nd its compnions to some specil mens nd some numericl qudrture rules, we refer the reder to the recent pper 3 by SS Drgomir nd S Wng Fink 4 used the integrl reminder from Tylor series expnsion to show tht for f n ) bsolutely continuous on, b, then the identity ) b 45) f t) dt n b ) f x) + F k x) + K F x, t) f n) t) dt n is shown to hold where 46) K F x, t) k x t)n n )! p x, t) n with p x, t) being given by t, t, x p x, t) t b, t x, b

225 7 P Cerone nd F k x) n k k! x ) k f k ) ) + ) k b x) k f k ) b) Fink then proceeds to obtin vriety of bounds from 45), 46) for f n) L p, b It my be noticed tht 45) is gin n identity tht involves function evlutions t three points to pproximte the integrl from the resulting inequlities See Mitrinović, Pečrić nd Fink 6, Chpter XV for further relted results In the current rticle split Peno kernel is utilised in which the brnches re products of two Appell-like polynomils, unlike Perce et l 7 who ssumed tht ech of the brnches themselves were Appell-like polynomils We shll focus, in this chpter, on the Peno kernel with only two brnches giving rise to rules involving evlution t three points Thus, the brnches of the Peno kernel re of product form The generl result then llows for gret flexibility in deriving integrtion rules which involve evlution t n interior point nd two end points Bounds re obtined in terms of the L p, b norms nd thus llowing, in dvnce, for the determintion of prtition required to chieve given error tolernce Simpson type formule re obtined in Section 43 nd perturbed rules using Grüss type inequlities involving the Chebychev functionl re investigted in Section 44 More perturbed results re obtined in Section 45 where the ide of bounds in terms of seminorms is utilised These cn in turn be bounded in terms of Lebesgue norms by ssuming stronger conditions on the function nd/or the Peno kernel It should be noted tht Perce et l 7 cll the Appell-like polynomils by the term hrmonic polynomils 4 Fundmentl Results The following lemm is prmount to the development of the subsequent work see lso Cerone 3) Lemm 4 Let r k, s k, u k, v k H for k N be sequences of polynomils which re such tht w k H if 47) w k t) w k t), w 0 t), t R Further, define K n x, t), p n t) nd q n t) by p n t) r n m t) s m t), t, x 48) K n x, t) q n t) u n m t) v m t), t x, b Then for f :, b R nd f n ) bsolutely continuous on, b, the identity 49) ) n K n x, t) f n) t) dt n m) +q k) n n f t) dt + k0 ) k { p k) n b) f n k) b) p k) n ) f n k) ) x) q n k) x) f n k) x) }

226 4 PRODUCT BRANCHES OF PEANO KERNELS AND NUMERICAL INTEGRATION 8 holds, where 40) with p k) n ) U ) k j r n m j ) s m k+j ), jl q n k) ) U ) k j u n m j ) v m k+j ) jl 4) U min {k, n m}, L mx {0, k m} Proof Let 4) J n, x, b) k0 K n x, t) f n) t) dt Then integrtion by prts from utilising 48) gives n 43) J n, x, x) ) n+k x x p k) n t) f n k) t) + ) n j0 p n) n t) f t) dt Now, using the Leibniz rule for differentition of product gives from 48) k ) k d p k) j r n m t) d k j s m t) n t) j dt j dt k j It should be noted from 47) tht 44) w j) k t) jmx{0,k m} w k j t), j k 0, j > k nd so from 43), since r n m t) nd s m t) re polynomils stisfying 47) nd using 44) gives min{k,n m} ) k 45) p k) n t) r n m j t) s m k+j t) j Similrly, from 4) 46) where J n x, x, b) 47) q k) n t) n ) n+k q n k) k0 min{k,n m} jmx{0,k m} + ) n x t) f n k) t) q n n) t) f t) dt, ) k u n m j t) v m k+j t) j Further, from 45) nd 47) we deduce tht j n m since the subscripts of u nd v re nonnegtive nd thus ) n n 48) p n) n t) q n n) t) n m m) b x

227 9 P Cerone Combining 4), 43) nd 46) on using 48) redily produces the identity 49) where 40) is obtined from 45) nd 47) Remrk 4 If we llow w k P provided 49) w k t) ξ k w k t), w 0 t), t R, then n ppropritely modified Lemm 4 would still hold When ξ k k, then such functions stisfying 49) were defined by Appell in 880, nd re now known s Appell polynomils see, for exmple, 0 for n extensive tretment of relted results) For ξ k, tht is stisfying 47), Perce et l 7 cll them hrmonic polynomils Polynomils stisfying 49) will be termed Appell-like polynomils Theorem 4 Let f :, b R be mpping such tht f n ) is bsolutely continuous on, b The following inequlities hold for f n) L p, b, p nd for ll x, b, n ) τ n x) : n { 40) f t) dt ) k+ p k) n x) m k0 q n k) x) f n k) x) + q n k) b) f n k) b) p k) n ) f )} n k) f n) B n, x) for f n) L, b ; where 4) 4) with f n) p B n q, x) q for f n) L p, b, with p >, p + q ; f n) θ n x) for f n) L, b, B n q, x) x p n t) q dt + θ n x) M n, x) + M n x, b) x q n t) q dt M n, x) sup p n t), M n x, b) sup q n t), t,x t x,b p n t), q n t) re defined by 48), p k) n t), q k) n t) re s given by 40), nd the Lebesgue norms re defined by f n) : ess sup f n) p : t,b + M n x, b) M n, x), f n) t) <, f n) t) p dt ) p, p Proof Tking the modulus of 49) gives 43) τ n x) J n, x, b), where τ n x) is s defined by the left hnd side of 40) nd J n, x, b) by 4)

228 4 PRODUCT BRANCHES OF PEANO KERNELS AND NUMERICAL INTEGRATION 0 Now, observe tht, from 4), 44) J n, x, b) f n) K n x, t) dt, where from 48) 45) K n x, t) dt x p n t) dt + x q n t) dt Thus, combining 43), 44) nd 45) gives the first inequlity in 40) Further, using Hölder s integrl inequlity, we hve the result, from 4), J n, x, b) f n) p K n x, t) q dt ) q, p +, p >, q which, upon using 48) produces the second inequlity in 40) from 43) Finlly, observe tht J n, x, b) f n) sup t,b K n x, t), where, from 4) sup K n x, t) mx t,b nd so using the well known identity { sup p n t), sup q n t) t,x t x,b } 46) mx {X, Y } X + Y + Y X produces, from 43), the third identity nd the theorem is completely proved Remrk 4 Lemm 4 recptures n identity obtined by Perce et l 8 if both p n t) nd q n t) stisfy 47) However, here p n t) nd q n t) do not in themselves stisfy 47) but re the product of two polynomils tht do Although in some sense the following result is specilistion of Theorem 4, it is deemed to be of such importnce, recpturing number of pst works s prticulr cses, tht it is itself referred to s Theorem in its own right Theorem 43 Let f :, b R be mpping such tht f n ) is bsolutely continuous on, b The following inequlities hold for f n) L p, b, p nd

229 P Cerone for α, x, β, b with α, x, β b, 47) where 48) τ C n x) : q k) n x) f n) n! f n) p n! B n C q, x) f t) dt n n m) k0 { ) k+ p k) n x) f n k) x) + q n k) b) f n k) b) p k) n ) f )} n k) B C n, x) for f n) L, b ; BC n q, x) q for f n) L p, b, with p >, p + q ; f n) n! θc n x) for f n) L, b, α ) nq+ B n m) q +, mq + ) ) + x α) nq+ Ψ mq, n m) q; α x α) + β x) nq+ Ψ mq, n m) q; b β β x + b β) nq+ B mq +, n m) q + ), α < x < β b x ) nq+ χ n m) q, mq, α + b x) nq+ χ x ) n m) q, mq, b β β x ), β < x < α b 49) θc n x) { mx A n m A + C A m, B n m B + D B } m, α < x < β b mx {A n m A C) m, B n m B D) m }, β x α b 430) nd p k) n t) U jl q n k) t) U jl ) k t ) n m j j n m j)! t α)m k+j m k+j)! k j ) t b) n m j n m j)! t β)m k+j m k+j)! with U nd L s given by 4), Ψ j, k; X) 0 uj X + u) k du, 43) χ j, k; X) 0 uj X u) k du, B j, k) χ j, k ; ) is the Euler bet function, nd 43) A x, C x α, B b x, D β x

230 4 PRODUCT BRANCHES OF PEANO KERNELS AND NUMERICAL INTEGRATION Proof Let, for n, m N nd m n 433) nd rn m C t) t )n m n m)!, s C m t) t α)m m! u C n m t) t b)n m n m)!, vm C t) t β)m m!, where α, x, β b nd then, from 48), nd using the superscript C to identify the bove polynomils, which stisfy 47), giving 434) p C n t) t )n m n m)! t α)m m! nd q C n t) t b)n m n m)! t β)m m! Now, from 4) nd 434) 435) B C n q, x) n m) { x n!) q t ) n m)q t α mq dt } + b t) n m)q t β mq dt : B C n q, x) n!) q Now, for α, x) nd β x, b, then from 435) B C n q, x) α β + t ) n m)q α t) mq dt + b t) n m)q β t) mq dt + x α x β t ) n m)q t α) mq dt b t) n m)q t β) mq dt which upon substitution u t α, first prt of 48) in terms of 43) t x α, b t β x Similrly, for α x, b nd β, x then B C n q, x) x which on substituting u t x 48) Further, from 4) nd 43) t ) n m)q α t) mq dt + b t nd b x nd t β b β x respectively, produces the b t) n m)q t β) mq dt respectively gives the second result in 436) θ C n x) n m) n! mx { : θ C n x) n! } sup t ) n m t α m, sup b t) n m t β m t,x t x,b : { n! mx M C n, x), M } n C x, b)

231 3 P Cerone Now, from 436) nd M C n, x) x ) n m sup t,x t α m mx {α, x α} m, α, x, x ) n m α ) m, α x, b M C n x, b) b x) n m sup t x,b t β m mx {β x, b β} m, β x, b, b x) n m b β) m, β, x which upon using 46) produces 49), where A, B, C, D re s defined in 43) nd the theorem is proved Remrk 43 The results of Theorem 43 re quite generl, giving previously reported results s specil cses It is perhps best to exmine the Peno kernel 48) where p n t) nd q n t) re s given by 434), where the superscript C is used to denote tht the kernel involves coupled kernels resulting in function evlution of t most three points For m 0 the results of Cerone et l 8 involving n time differentible Ostrowski results re obtined Tking m n reproduces the results of Cerone nd Drgomir 7 involving three point results for n time differentible functions consisting of evlutions t the end points nd n interior point If m n nd α β x, then the generlised trpezoidl results for the n time differentible results of Cerone et l 9 re recovered If we tke m n nd α β x, then the results of Fink 4 re obtined within this quite generl frmework The following corollry gives more elegnt representtion of the results through prmetristion of the intervls under considertion through representtion of α nd β s convex combintion of the end points Corollry 44 Let the conditions of Theorem 43 hold Then n 437) τ C n x) : f t) dt ) k+ d k) n x) f n k) x) n kn m ) k+ Q k) n B C n,x) n! B C n q,x) q n! θ C n x) n! k0 b) f n k) b) P n k) ) f n k) ) f n), for f n) L, b ; f n) p, for f n) L p, b, with p >, p + q ; f n), for f n) L, b,

232 4 PRODUCT BRANCHES OF PEANO KERNELS AND NUMERICAL INTEGRATION 4 where 438) Bn C q, x) λ nq+ B n m) q +, mq + ) + ) A nq+ + B nq+ λ) nq+ λ Ψ mq, n m) q, λ, 0 λ < ; χ n m) q, mq, λ), λ, 439) θ C n x) b + x + b n + λ m, 0 λ < ; λ m, λ, with Ψ j, k; X), χ j, k; X) nd B j, k) s given by 43) nd A x, B b x Further, 440) nd d k) n x) A n k B) n k n ) m P n k) ) A n k λ) n k Q k) n b) B n k λ n k U k j λ) m k+j n m j)!m k+j)!, jl k n m n ), k n m m k n m ), k n m n m with, from 4), U min {k, n m} nd L mx {0, k m} Proof Tking α λx+ λ) nd β λx+ λ) b in Theorem 43 gives the bove results fter some simplifiction Remrk 44 Tking λ 0 in Corollry 44 implies tht α nd β b, producing n Ostrowski type result similr to those obtined in Cerone et l 8 Tking λ gives generlized trpezoidl results for n time differentible functions of Cerone et l 9 The inequlity 44) f t) dt b 6 43 Simpson Type Formule f ) + 4f + b ) + f b) b )5 f 4) 880 is well known s the Simpson inequlity where the mpping f :, b R is ssumed to hve f 4) bounded on, b) A review rticle by Drgomir et l presents bounds in terms of t most first derivtive while Pečrić nd Vrošnec 8 obtin bounds in terms of f k) L p, b, p nd k 4 The current work

233 5 P Cerone looks t recpturing the bove results s specil cses of the development presented in the previous section Let m in 434), let α nd β be shown to depend on x nd subscripted to denote their dependence on n Further, superscript of S is used to depict the reltionship with Simpson s rule Then 44) p S n t) with t )n n )! t α n x)), q S n t) t b)n n )! t β n x)), 443) α n x) λ n x + λ n ), β n x) λ n x + λ n ) b, λ n n 3 Corollry 45 Let the conditions of Corollry 44 hold, then τ S n x) n 444) : f t) dt ) k+ d k) n x) f n k) x) k0 ) n λ n B f b) + A f ) n f n), for f n) L, b ; where 445) B S n q, x) B S n,x) n! B S n q,x) q n! θ S n x) n! A nq+ + B nq+ f n) p, for f n) L p, b, with p >, p + q ; f n), for f n) L, b, λ nq+ n B n ) q +, q + ) + λ n ) nq+ Ψ q, n ) q, ) λ n λ n, 0 λ n < ; χ n ) q, q, λ n ), λ n, b 446) θ S n x) + x + b n + λ n, 0 λ n < ; λ n, λ n, 447) d k) n x) A n k B) n k n min{k,n } jmx{0,k } ) k λ n ) j n j)! k + j)!, with Ψ j, k; X), χ j, k; X) nd B j, k) s given by 43) nd A x, B b x, λ n x 3 Proof Tking p S n t), qn S t), α n x), β n x) nd λ n n 3 results fter some simplifiction produces the bove

234 4 PRODUCT BRANCHES OF PEANO KERNELS AND NUMERICAL INTEGRATION 6 If n,, 3 nd 4, then we obtin the Simpson type results of Drgomir et l nd Pečrić nd Vrošnec 8, provided tht x is tken t the midpoint, tht is, t x +b It my be seen tht dk) n x) 0 from 440) nd 447) when n k is even so tht only even derivtives re present when x +b Further, from 447), d ) 4 0 nd so there is no f x) term in the qudrture rule Tking, for exmple, n 4, q nd λ in 44) 447) reproduces 44) 44 Perturbed Results Perturbed versions of the results of the previous sections my be obtined by using Grüss type results involving the Chebychev functionl 448) T f, g) M fg) M f) M g) with 449) M f) b f t) dt For f, g :, b R nd integrble on, b, s is their product, then 450) T f, g) T f, f) T g, g), Drgomir for f, g L, b ; Γ γ T f, f), Mtić et l 5 for γ g t) Γ, t, b ; Γ γ)φ φ) 4 Grüss see 6, pp 95-30) φ f Φ, t, b Drgomir obtins numerous results if either f or g or both re known, lthough the first inequlity in 450) hs long history see for exmple 6, pp 95-30) The inequlities in 450) when proceeding from top to bottom re in the order of incresing corseness See lso Cerone for the perturbed results nd their implementtion s composite rules Theorem 46 Let the mpping f :, b R be such tht f n ) is bsolutely continuous Then the following inequlity holds Nmely, 45) τ n x) ) n U n x) S n f;, b) b ) κ n x) f n) b S n f;, b), f n) L, b, b ) κ n x) b ) Φ n x) φ n x)) 4 ) Γn γ n, γ n f n) t) Γ n, t, b, Γ n γ n ), φ n x) K n x, t) Φ n x), t, b, where τ n x) is s defined in 40), m 45) U n x) ) m k P n,m,k x) P n,m,k ) + Q n,m,k b) Q n,m,k x), k0

235 7 P Cerone 453) P n,m,k t) r n m k) t) s k t), Q n,m,k t) u n m k) t) v k t) with r ), s ), u ), v ) stisfying 47), 454) S n f;, b) f n) b) f n) ), b nd 455) κ n x) b with K n x, t) being s defined by 48) K n x, t) dt ) Un x), b Proof Associting f t) with ) n K n x, t) nd g t) with f n) t), then from 49), 448) nd 449), we obtin ) T ) n K n x, ), f n) ) ) ) M ) n K n x, ), f n) ) M ) n K n x, )) M f n) ) nd thus 456) ) b ) T ) n K n x, ), f n) ) τ n x) ) n f n ) b) f n ) ) ) U n x), b where τ n x) is the left hnd side of 40) nd 457) U n x) Now, using 48) x x p n t) dt p n t) dt + x x q n t) dt r n m t) s m t) dt from which repeted integrtion by prts nd using the fct tht r ) nd s ) stisfy 47) gives on using 453) 458) x p n t) dt m ) m k P n,m,k t) x k0 A similr rgument gives on using 456) 459) q n t) dt m ) m k Q n,m,k t) b x k0 nd so combining 458) nd 459) in 456) gives 45) Now for the bounds in 45)

236 4 PRODUCT BRANCHES OF PEANO KERNELS AND NUMERICAL INTEGRATION 8 From 448) we hve, for K n x, t) s defined by 48), 460) T ) n K n x, ), ) n K n x, )) M Kn x, ) ) M K n x, )) b K n x, t) dt b b κ n x), s defined in 455) since U n x) K n x, t) dt Further, using 448), 45) nd 454) gives T ) f n) t), f n) t) K n x, t) dt ) ) ) M f n) t) M f n) t) b { b f n) t) dt f n) t) dt b f n) } S n f;, b) nd so combining the bove result with 460) produces the first inequlity For f n) L, b L, b with strict inclusion) then ) 46) 0 T f n) t), f n) t) b f n) t) dt f n) t) dt b b ) Γn γ n, where γ n f n) t) Γ n, t, b, nd the third inequlity in 45), is due to Grüss Hence the second inequlity in 45) is obtined which is corser thn the first Further, from 460) ) 0 κ Φn x) φ n n x), where φ n x) K n x, t) Φ n x), t, b, which, when combined with 46), produces the third bound in 45) which is the corsest bound of ll The proof of the theorem is thus complete Remrk 45 The second result in 45) is generlistion of result by Perce et l 7 in which ech of the brnches of the Peno kernel 48) stisfy 47) Tht is, P n t), t, x ; 46) Kn x, t) Q n t), t x, b,

237 9 P Cerone where P n t) nd Q n t) stisfy 48) With the Peno kernel 46), they obtined the result Theorem 47 Assume tht f :, b R is such tht f n) is integrble nd γ n f n) Γ n for ll t, b Put 463) U n x) : b Q n+ b) Q n+ x) + P n+ x) P n+ ) Then for ll x, b, we hve the inequlity n 464) f t) dt ) k+ Q k b) f k ) b) + P k x) Q k x)) f k ) x) k P k ) f k ) ) ) n U n x) f n ) b) f ) n ) where { 465) K x) : b x P n t) dt + x K x) Γ n γ n ) b ), Q n t) dt U n x) } In the recent pper, Drgomir proved the following refinement of 464) Theorem 48 Assume tht the mpping f :, b R is such tht f n ) is bsolutely continuous on, b nd f n) L, b n ) If we denote f n ) ;, b : f n ) b) f n ) ), b then we hve the inequlity n 466) f t) dt ) k+ Q k b) f k ) b) + P k x) Q k x)) f k ) x) k P k ) f k ) ) ) n Q n+ b) Q n+ x) + P n+ x) P n+ ) f n ) ;, b K x) b ) b for ll x, b nd K x) s is given in 465) f n) ) f n) ;, b ) K x) b ) Γ n γ n ) if f n) L, b), The results of Theorem 46 re generlistions of these which llow ech of the brnches themselves to be mde up of products of functions stisfying 47)

238 4 PRODUCT BRANCHES OF PEANO KERNELS AND NUMERICAL INTEGRATION 30 Corollry 49 Let the conditions on f of Theorem 46 hold nd let α, x, β, b with α, x < β b then, n τ m) C n x) ) n Un C 467) x) S n f;, b) b ) κ C n x) f n) b S n f;, b), f n) L, b, b ) κ C n x) ) Γn γ n, γ n f n) t) Γ n, t, b, ) Φ C n x) φ C n x) Γ n γ n ) b ), 4 φ C n x) κ C n x, t) Φ C n x), t, b, where τ C n x) nd S n f;, b) re s defined by 47) nd 454) respectively, p C t )n m t α)m n t), t, x n m)! m! 468) Kn C x, t) 469) U C n x) with nd 470) nd B n, m) B b n, m) B j, k, X) κ C n x) { q C n t) t b)n m n m)! t β)m, t x, b, m! B n, m) + n m)!m! B b n, m), x ) n+, n + α ) m α ) n+ B n m +, m +, x ), α α ) n b x) n+, n + β b ) ) n m b β) n+ B n m +, m +, b x b β, β b X 0 n m)!m! b ) 47) φ C n x) min u j u) k du, the incomplete bet function, B n, m) + B b n, m) ) U C } n x) b { } φ n x), φ b n x), Φ C n x) mx { Φ n x), Φ b n x) },

239 3 P Cerone with φ n x) inf t,x pc n t), φ b n x) inf t x,b qc n t), Φ n x) sup p C n t), Φ b n x) sup qn C t) t,x t x,b Proof Let r C ), s C ), u C ) nd v C ) be s defined in 433) giving from 434) nd 48) K C n x, t) s shown in 468) Now, 47) U C n x) nd so using 468) consider 473) n m)!m! x K C n x, t) dt p C n t) dt If α, let α ) u t, then x Similrly, 474) x x p C n t) dt + x q C n t) dt t ) n m t α) m dt x )n+, α n + t ) n m t α) m dt ) n α ) n+ x α n m)!m! x q C n t) dt If β b, let b β) v b t to give x x 0 u n m u) m du t b) n m t β) m dt )n b x) n+, β b n + t b) n m t β) m dt ) n m b β) n+ x b β Substitution of the bove results into 47) gives 469) 0 v n m v) m dv Further, from 468), K C n x, t) { x dt t ) n m) t α) m dt n m)!m! } + x t b) n m) t β) m dt which, on following similr procedure to the bove gives 475) K C n x, t) dt n m)!m! Hence, since 476) κ C n x) b B n, m) + B b n, m) K C n x, t) ) U C dt n x), b,

240 4 PRODUCT BRANCHES OF PEANO KERNELS AND NUMERICAL INTEGRATION 3 then from 475) nd 469) we obtin 470) For the first inequlity we observe tht, from 468), { Φ C n x) sup Kn C x, t) mx t,b sup p C n t), sup qn C t) t,x t x,b nd { } φ C n x) inf t,b KC n x, t) min inf t,x pc n t), inf t x,b qc n t) The corollry is thus completely proven where the inequlities re in incresing corseness s discussed in Theorem 46 Remrk 46 If α nd β b, then perturbed generlised Ostrowski type result is obtined If α β x, perturbed trpezoidl type results re obtined for n-time differentible functions Corollry 40 Let the conditions of Theorem 46 hold nd let α, x, β, b with α, x, β b, then n τ m) C n x) ) n 477) Un C x) S n f;, b) b ) κ C n x) f n) b S n f;, b), f n) L, b, where τ C n 478) b ) κ C n b ) K C n x, t) x) ) Γn γ n, γ n f n) t) Γ n, t, b, ) Φ C n x) φ C n x) Γ n γ n ) φ C n 4 x) Kn C, x, t) Φ C n x), t, b, x) nd S n f;, b) re s defined by 437) nd 454) respectively, p C n q C n t) t) t )n m n m)! t b)n m n m)! } t xλ + λ) ))m, t, x m! t xλ + λ) b))m, t x, b, m! 479) Un C x) B n, m) x ) n+ + ) n b x) n+, n m)!m! with nd B n, m) B j, k, X) X 0 n +, λ 0, ) m λ n+ B n m +, m +, λ), λ 0, u j u) k du, the incomplete bet function,

241 33 P Cerone 480) κ C n x) { λ n+ B n, m) n m)!m! b ) x ) n+ nd { φ C n x) min { Φ C n x) mx + b x) n+ U C n x) b inf t,x pc n t), inf t x,b qc n ) } } t), sup p C n t), sup qn C t) t,x t x,b } Proof Tking α λx + λ) nd β λx + λ) b in Corollry 49 produces the bove results fter some simplifiction Remrk 47 Tking λ 0 in Corollry 40 implying tht α nd β b produces perturbed Ostrowski type results If λ, then perturbed trpezoidl type results re obtined for n time differentible functions Corollry 4 Let f :, b R be mpping such tht f n ) is bsolutely continuous on, b Then, for α, x, β, b with α, x, β b, the perturbed Simpson rule τ S n x) )n n U 48) n S x) S n f;, b) b ) κ S n x) n f n) b S n f;, b), f n) L, b, ) b ) κ S Γn γ n x) n, γ n n f n) t) Γ n, t, b, b ) n ) Φ S n x) φ S n x) Γ n γ n ), 4 φ S n x) Kn S x, t) Φ S n x), t, b, where τ S n x) nd S n f;, b) re s defined by 445) nd 454) respectively, p S t )n n t) t α n x)), n )! 48) Kn S x, t) qn S t b)n t) t β n )! n x)), with 483) α n x) λ n x + λ n ), β n x) λ n x + λ n ) b, λ n n 3, 484) U S n x) n λ n) λ n n + )! x ) n+ + ) n b x) n+,

242 4 PRODUCT BRANCHES OF PEANO KERNELS AND NUMERICAL INTEGRATION ) { λ n+ κ S B n, 3, λ n x) n )! b ) ) x ) n+ + b x) n+ ) U S } n x) b nd 486) with φ S n x) Φ S n x) { } n )! min inf X U), inf Y V ), U 0,x V 0,b x { } n )! min X U), sup Y V ), sup U 0,x V 0,b x X U) U n U λ n x )), Y V ) ) n V n V λ n b x)) Proof Tking m in Corollry 40 nd 469) nd 483) in plce of 478) gives 487) U S n x) x p S n t) dt + x q S n t) dt Tking m nd λ λ S n in 479) gives Un S x) Alterntively, direct clcultion gives, Similrly, x p S n t) dt x x q S n t) dt t ) n n )! t α n x)) dt t )n n + )! n + ) t α n x)) t ) x )n n + )! n + ) x α n x)) x ) x )n n + )! n + ) λ n) x )n n + )! n λ n) λ n x t b) n n )! t β n x)) dt ) n b x) n n + )! n) λ n) λ n Combining the bove results into 437) produces 484) Now, to determine κ S n x) from 476) we hve, using 48) b K S b n x, t) x dt p S b n t) ) dt + q S n t) ) dt x x t

243 35 P Cerone For direct clcultion we require integrtion by prts twice Alterntively, utilising 470) with m nd α n x), β n x) being s given by 483) produces the stted result 485) Now, nd on using 483), sup p S n t) t,x sup qn S t) t x,b Φ S n x) mx { sup p S n t), sup qn S t) t,x t x,b sup U n U λ n x )), n )! U 0,x sup ) n V n V λ n b x)) n )! V 0,b x) In similr fshion { } φ S n x) min inf t,x ps n t), inf t x,b qs n t), nd using the bove expressions with sup replced by inf gives the results s stted in 486) The proof is now complete Remrk 48 The perturbed results obtined bove through the use of the Chebychev functionl 448) nd the resulting bounds given by 450) my be dvntgeous when compred to the first bounds in 40), 47), 436) nd 444) For functions g, h :, b R nd γ g t) Γ, then Γ γ g It is however, difficult to compre h g obtined for the unperturbed results of previous sections, with the perturbed bounds of the form where b ) h σ g) < b ) h Γ γ < b ) σ g) b g S g;, b), S g;, b) nd φ h t) Φ }, Φ φ) Γ γ), g ) g b) b This is so since comprison between h nd h cnnot redily be mde 45 More Perturbed Results Using Seminorms In recent rticle Cerone nd Drgomir 5 obtined the following results of Grüss type for the Chebychev functionl T f, g) They utilised the notion of seminorm introduced by Cerone nd Drgomir 4 where f p : b f s) f t) p p dsdt), for f L p, b, p, ), 488) nd f : ess sup f s) f t), for f L, b s,t),b

244 4 PRODUCT BRANCHES OF PEANO KERNELS AND NUMERICAL INTEGRATION 36 If we consider f :, b R, where f s, t) f s) f t), then 489) f p f p, p, with p being the usul Lebesgue p norms on, b Using the properties of the Lebesque p norms, we my deduce the following seminorm properties for p : i) f p 0 for f L p, b nd f p 0 implies tht f c c is constnt) e in, b ; ii) f + g p f p + g p if f, g L p, b ; iii) λf p λ f p We note tht if p, then, f f t) f s)) dtds ) ) b b ) f f t) dt The following theorem giving bounds for the Chebychev functionl in terms of seminorms 488) holds see lso Cerone nd Drgomir 5) Theorem 4 Let f, g :, b R be mesurble on, b Then the inequlity 490) T f, g) b ) f p g q holds provided the integrls exist, where T f, g) is the Chebychev functionl given by 448) 449), p, q or q, p or p >, p + q, nd f is defined by 488) Proof Using Korkine s identity, we hve T f, g) b ) f x) f y)) g x) g y)) dxdy, where T f, g) is the Chebychev functionl defined by 448) Now, if f L, b, then T f, g) b ) f x) f y) g x) g y) dxdy ess sup f x) f y)) b ) x,y),b b ) f g, nd the inequlity is proved for p, q g x) g y) dxdy

245 37 P Cerone A similr rgument pplies for p, q If p >, p + q, then pplying Holder s integrl inequlity for double integrls, we deduce tht T f, g) b ) b ) b ) f p g q f x) f y) g x) g y) dxdy f x) f y) p dxdy ) p g x) g y) q dxdy ) q nd the theorem is proved Using the fct tht if f :, b R is bsolutely continuous then f s) f t) s t f u) du, the following result ws obtined by Cerone nd Drgomir 4 nd the proof is presented here for the ske of completeness Theorem 43 For f :, b R bsolutely continuous on, b the following inequlities hold i) If p, ), then + p b ) p f p + ) p + ), f L, b, p 49) f p δ ) p b ) δ + p f p + δ) p + δ) γ, f L γ, b, p γ >, γ + δ b ) p f, ii) b ) f, f L, b ; 49) f b ) δ f γ, f L γ, b, γ >, γ + δ ; f

246 4 PRODUCT BRANCHES OF PEANO KERNELS AND NUMERICAL INTEGRATION 38 Proof As f :, b R is bsolutely continuous, then f s) f t) s t f u) du for ll s, t, b, nd then 493) f s) f t) s t f if f L, b ; s f u) du s t δ f γ if f L γ, b, γ >, γ + δ ; t f if f L, b nd so for p, ), we my write nd then from 489) f s) f t) p s t p f p if f L, b ; s t p δ f p γ if f L γ, b, γ >, γ + δ ; f p if f L, b, f s t p dsdt) p if f L, b ; 494) f p f γ f s t p δ dsdt) p dsdt ) p if f L α, b, γ >, γ + δ ; if f L, b Further, since s t p dsdt ) p b s s t) p dt + s t s) p dt b s ) p+ + b s) p+ ds p + ) ) p ds p giving nd s t p β dsdt + p b ) p, p + ) p + ) p dsdt ) p ) p we obtin, from 494), the stted result 49) δ ) p b ) δ + p, p + δ) p + δ) p b ) p,

247 39 P Cerone Using 493) we hve for p ) tht f f ess sup s t s,t),b f γ ess f sup s,t),b s t δ b ) f b ) δ f γ f nd the inequlity 49) is lso proved Remrk 49 If p q is tken in Theorem 4 then the first result in 450) is obtined Theorem 44 Let the mpping f :, b R be such tht f n ) is bsolutely continuous, then the following inequlity holds, provided the integrls exist Nmely, 495) τ n x) ) n U n x) S n f;, b) b ) K n x, ) q f n) ), p >, p p + q, where K n x, t) is Peno kernel defined by 48), U n x) K n x, t) dt nd S n f;, b) is s defined in 454) Proof Following the proof of Theorem 46 nd ssociting f t) with ) n K n x, t) nd g t) with f n) t), then from 49), 448), 449) we obtin the result 454) giving b ) T ) n K n x, ), f n) ) ), the left hnd side of 495) Now for the bound we hve from 490) the inequlity s given Bounds my be obtined tht re more esily clculted thn those in 495) by plcing stronger conditions on K n x, ) nd/or f n) ) The following corollry ssumes tht f n) is bsolutely continuous Corollry 45 Let the mpping f :, b R be such tht f n) is bsolutely continuous, then 496) τ n x) ) n U n x) S n f;, b)

248 4 PRODUCT BRANCHES OF PEANO KERNELS AND NUMERICAL INTEGRATION 40 K n x, ) b ) δ K n x, ) b ) K n x, ) p b ) p p + ) p + ) p p δ ) p b ) δ + p p+δ)p+δ) p f n+), f n+) L, b ; f n+) γ, f n+) L γ, b, γ >, γ + δ ; f n+) ; K n x, ) q K n x, ) q f n+), f n+) L, b, p >, p + q ; f n+) γ, f n+) L γ, b, γ >, γ + δ, p >, p + q ; b ) p K n x, ) q f n+), f n+) L, b, p >, p + q, b ) K n x, ) f n+) 6, f n+) L, b ; δ b ) δ + δ + ) δ + ) K n x, ) f n+) γ, f n+) L γ, b, γ >, γ + δ ; b ) K n x, ) f n+) Proof Using result 495) of Theorem 44, we hve tht f n) ) is bsolutely continuous from the ssumptions of the current theorem nd so the conditions of Theorem 43 hold for f n) ) Remrk 40 If we choose K n x, t) from 48) such tht K n x, t) is bsolutely continuous, then the results of Theorem 43 would hold nd K n x, ) could be determined in terms of Knx,t) t L, b, where the differentition is tken over ech subintervl This would result in bounds from 496) involving 7 brnches Thus, with the ssumption of bsolute continuity of K n x, ) then, using Theorem 43 we would hve from 49) nd 49), where it is understood tht K n x, ) represents differentition over ech subintervl, x nd x, b, i) for q, ), K n x, ) q nd q b ) + q K n x, ), K n x, ) L, b ; η ) p b ) η + q K n x, ) ν, K n x, ) L ν, b, ν >, ν + η ; b ) q K n x, ),

249 4 P Cerone ii) b ) K n x, ), K n x, ) L, b ; K n x, ) b ) η K n x, ) ν, K n x, ) L ν, b, ν >, ν + η ; K n x, ) Remrk 4 For K n x, t) to be bsolutely continuous, it is sufficient to hve tht, from 48), p n x) q n x) As n exmple, consider the Simpson type kernel s given by 48), then using 483) gives p S x )n n x) n )! λ n) nd qn S x) ) n b x) n n )! λ n) Thus, the bove condition is stisfied for x +b Now, nd so K S n t K S n t x, t) ) + b, t from which Kn S t the detils t ) n n )! t b) n n )! t ) n n )! n t α n x)) + λ n x ), t, x n t β n x)) + λ n b x), t x, b ) b n t ) n ) λ n, t, + b ) b + b n t b) n ) λ n, t, b t b) n n )! t) + b ν,, ν, my be obtined explicitly We omit 46 Concluding Remrks Qudrture rules eminting from Peno kernels involving brnches of products of Appell-like polynomils stisfying 47) hve been investigted in this chpter The rules involve function evlutions t the boundry points nd n interior point Explicit priori bounds re obtined in terms of vriety of norms so s to llow the possibility of determining the prtition required to chieve prescribed error tolernce Perturbed rules re exmined nd bounds re obtined through detiled study of the Chebychev functionl The work sets the foundtion for further investigtion of this new clss of qudrture rules which includes gret del of work involving three point rules s specil cses

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251 Bibliogrphy P APPELL, Sur une clsse de polnômes, Ann Sci Ecole Norm Sup, 9) 880), 9-44 P CERONE, Perturbed rules in numericl integrtion from product brnched Peno kernels, submitted, RGMIA Rep Coll, 3 4) 000), Article 0 3 P CERONE, Three point rules in numericl integrtion, Journl of Nonliner Anlysis, ccepted) 4 P CERONE nd S S DRAGOMIR, New bounds for perturbed generlised Tylor s formul, Mth Ineq & Applics, submitted) 5 P CERONE nd S S DRAGOMIR, Some bounds in terms of seminorms for Ostrowski- Grüss type inequlities, Soochow J Mth, submitted) 6 P CERONE nd S S DRAGOMIR, Three point qudrture rules involving, t most, first derivtive, submitted, RGMIA Res Rep Coll, 4) 999), Article 8 7 P CERONE nd S S DRAGOMIR, Three point identities nd inequlities for n time differentible functions, RGMIA Res Rep Coll, 7) 999), Article 3 8 P CERONE, S S DRAGOMIR nd J ROUMELIOTIS, Some Ostrowski type inequlities for n-time differentible mppings nd pplictions, Demonstrtio Mth 9 P CERONE, S S DRAGOMIR, J ROUMELIOTIS, nd J SUNDE, A new generlistion of the trpezoid formul for n time differentible mppings nd pplictions, ccepted), Demonstrtio Mth 0 A DI BUCCHIANICO nd D LOEB, A Selected Survey of Umbrl Clculus ONLINE S S DRAGOMIR, Better bounds in some Ostrowski-Grüss type inequlities, Tmkng J of Mth, ccepted) S S DRAGOMIR, RP ARGARWAL nd P CERONE, On Simpson s inequlity nd pplictions, J of Ineq & Appl, in press) 3 S S DRAGOMIR nd S WANG, Applictions for Ostrowski s inequlity to the estimtion of error bounds for some specil mens nd some numericl qudrture rules, Appl Mth Lett, 998), AM FINK, Bounds on the derivtion of function from its verges, Czech Mth Journl, 4 7) 99), M MATIĆ, JE PEČARIĆ nd N UJEVIĆ, On New estimtion of the reminder in Generlised Tylor s Formul, MthIneq & App, 3) 999), DS MITRINOVIĆ, JE PEČARIĆ nd AM FINK, Inequlities for Functions nd Their Integrls nd Derivtives, Kluwer Acdemic Publishers, Dordrecht, CEM PEARCE, J PEČARIĆ, N UJEVIĆ nd S VAROŠANEC, Generlistions of some inequlities of Ostrowski-Grüss type, Mth Ineq & Appl, 3 ), 000), J PEČARIĆ nd S VAROŠANEC, On Simpson s formul for Functions whose derivtives belong to L p spces, submitted Submitted rticles my be found t 43

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253 CHAPTER 5 Ostrowski Type Inequlities for Multiple Integrls by NS BARNETT, P CERONE nd SS DRAGOMIR 5 Introduction In 938, A Ostrowski proved the following integrl inequlity 9, p 468 Theorem 5 Let f :, b R be differentible mpping on, b) whose derivtive f :, b) R is bounded on, b), ie, f : f t) <, then we hve the inequlity 5) f x) b f t) dt for ll x, b The constnt 4 is the best possible 4 + x +b b ) sup t,b) ) b ) f For some generliztions of this result see 9, p by Mitrinović, Pe crić nd Fink Recent results on Ostrowski s inequlity my be found online t: In 975, GN Milovnović generlized Theorem 5 to the cse where f is function of severl vribles Following 8, let D {x,, x m ) i < x i < b i closure of D i,, m)} nd let D be the We now propose nd prove the following generlistion of Theorem 5 45

254 5 OSTROWSKI TYPE INEQUALITIES FOR MULTIPLE INTEGRALS 46 Theorem 5 Let f : R m R be differentible function defined on D nd let f x Mi M i > 0; i,, m) in D Then, for every X x,, x m ) D, b m 5) f X) m f y,, y m ) dy dy m b i i ) m i ) m 4 + xi i+bi b i i ) b i i ) M i i Ac- Proof Let X x,, x m ) nd Y y,, y m ) X D, Y D ) cording to Tylor s formul, we hve m f C) 53) f X) f Y ) x i y i ), x i where C y + θ x y ),, y m + θ x m y m )) 0 < θ < ) i Integrting 53), we obtin m 54) f X) mes D f Y ) dy D i where dy dy dy m nd mes D m i b i i ) From 54), it follows tht, f X) mes D f Y ) dy nd 55) D f X) mes D D m i m i D D D m f Y ) dy M i i f C) x i y i ) dy, x i f C) x i y i ) dy x i f C) x i x i y i dy, D x i y i dy, respectively, owing to the ssumption f x i Mi M i > 0; i,, m) Since we hve D i i x i y i dy i 4 b i i ) + x i ) i + b i, x i y i dy i mes D i x i y i dy i b i i i mes D) b i i ) 4 + xi i+bi b i i ) )

255 47 NS Brnett, P Cerone nd SS Drgomir Since mes D > 0, inequlity 55) becomes f x,, x m ) m b i i ) D i ) m 4 + xi i+bi b i i ) b i i ) M i, i nd the result is proved f Y ) dy Theorem 5 cn be generlised s follows see 8) Theorem 53 Let f : R m R be differentible function defined on D nd f x i Mi M i > 0; i,, m) in D Furthermore, let function X p X) be defined, integrble nd p X) > 0 for every X D Then, for every X D, f X) D m M p Y ) f Y ) dy D i D p Y ) x i y i dy p Y ) dy i D p Y ) dy This theorem cn be proved similrly to Theorem 5 For Theorem 54, we use the following nottion: m, n i N i,, m) ; 0 i0 < i < < ini i,, m) ; iki x iki iki, λ iki iki iki k i,, n i ; i,, m) k k,, k m ), X x,, x m ), X k x k,, x mkm ) ; D {X 0 < x i < ; i,, m} ; D k) { X k iki < x iki < iki k i,, n i ; i,, m) } ; dx dx dx m ; E f; k) f X k ) m f X) dx λ iki i Theorem 54 Let f : R m R be differentible function defined on D nd f x Mi M i > 0; i,, m) in D Then n i n m 56) f X) dx λ k λ mkm f X k ) D k k m m ni ) M i H x iki ; k i ), where i k Dk) H t; k i ) t iki ) + iki t)

256 5 OSTROWSKI TYPE INEQUALITIES FOR MULTIPLE INTEGRALS 48 Proof According to Theorem 5, we hve 57) E f; k) m M i H x iki ; k i ) λ iki Since n k n m k m D k) D, we hve i n i n m f X) dx λ k λ mkm f X k ) D k k m n i n m ni λ k λ mkm E f; k) n m λ k λ mkm E f; k) k k m k k m Using 57), the lst inequlity becomes n i n m f X) dx λ k λ mkm f X k ) D k k m n i n m m ) M i λ k λ mkm H x iki ; k i ) λ k k m i iki m ni ) M i H x iki ; k i ) i The proof is thus completed k Corollry 55 If x iki iki or x iki iki, from 54) it follows tht n i n m f X) dx λ k λ mkm f X k ) m ni ) M i λ ik i D k k m Furthermore, if λ iki n i holds, then f X) dx n n m D n i k n m k m i f X k ) Corollry 56 If x iki ik i + iki ), holds n i n m f X) dx λ k λ mkm f X k ) 4 D k k m Furthermore, if λ iki n i, we hve f X) dx n n m D The following result cn be found in 8 n i k n m k m m i f X k ) 4 m i k M i n i ni M i Theorem 57 Let f : R m R be differentible function defined on D {x,, x m ) i x i b i i,, m)} m i k M i n i λ ik i )

257 49 NS Brnett, P Cerone nd SS Drgomir nd let f x i Mi M i > 0, i,, m) in D Furthermore, let function x p x) be integrble nd p x) > 0 for every x D Then for every x D, we hve the inequlity: m M 58) f x) p y) f y) dy i D p y) x i y i dy D p y) dy i D p y) dy D 5 An Ostrowski Type Inequlity for Double Integrls 5 Some Inequlities in Terms of Norm The following inequlity of Ostrowski s type for mppings of two vribles holds see lso ): Theorem 58 Let f :, b c, d R be so tht f, ) is continuous on, b c, d If f x,y f x y exists on, b c, d nd is in L, b c, d), ie, f s,t : ess sup f x, y) x y < x,y),b) c,d) then we hve the inequlity: d 59) f s, t) dtds b ) c d c) b ) f x, y) 4 b ) + x + b for ll x, y), b c, d d c f x, t) dt + d c) ) 4 d c) + f s, y) ds y c + d ) f s,t 50) Proof We hve the equlity: x y x c s ) t c) f s,t s, t) dtds s ) f s s, y) y c) f s s, t) dtds c x y x ) y c) s ) f s s, y) ds s ) f s s, t) ds dt c x y c) x ) f x, y) f s, y) ds y x x ) f x, t) f s, t) ds dt c y y c) x ) f x, y) y c) x ) y c f x, t) dt + x x y c f s, y) ds f s, t) dtds

258 5 OSTROWSKI TYPE INEQUALITIES FOR MULTIPLE INTEGRALS 50 By similr computtions we hve, 5) x d x y d y) s ) t d) f s,t s, t) dtds s ) x d y) f s s, y) s ) f s s, y) ds d y d y f s s, t) dt x ds ) s ) f s s, t) ds dt d y) x ) f x, y) d y x ) f x, t) x x x ) d y) f x, y) d y) x ) d y f x, t) dt + f s, y) ds f s, t) ds dt x x y c f s, y) ds f s, t) dtds Now, 5) d x x y s b) t d) f s,t s, t) dtds s b) d y) x d y) f s s, y) s b) f s s, y) ds d y) b x) f x, y) d b x) f x, t) y x x d d y) b x) f x, y) d y) b x) d f x, t) dt + y d y f s s, t) dt ds ) b s b) f s s, t) ds dt x f s, y) ds f s, t) ds d y x y x dt f s, y) ds f s, t) dtds

259 5 NS Brnett, P Cerone nd SS Drgomir nd finlly 53) y x x c y c) y c) s b) t c) f s,t s, t) dtds y y c)f s s, y) s b) y c x s b) f s s, y) ds b x) f x, y) b x) f x, t) y c) b x) f x, y) y c) b x) y c f x, t) dt + x x c y f s s, t) dt c x f s, y) ds f s, t) ds x y x c dt f s, y) ds f s, t) dtds ds s b) f s s, t) ds If we dd the equlities 50) 53) we get in the right membership: y c) x ) + x ) d y) + d y) b x) + y c) b x) f x, y) d c) b ) + d x y x d y f s, y) ds d c) f x, t) dt + f s, t) dtds + x y c y d c) b ) f x, y) d c) b ) c f x, t) dt + x c d c x f s, y) ds b ) f s, t) dtds + f s, t) dtds f s, y) ds f s, t) dtds x d Define the kernels: p :, b R, q : c, d R given by: s if s, x p x, s) : s b if s x, b nd t c if t c, y q y, t) : t d if t y, d y y c ) dt f x, t) dt f s, t) dtds Now, using these, we deduce tht the left hnd side of this sum cn be represented s : d c p x, s) q y, t) f s,t s, t) dtds

260 5 OSTROWSKI TYPE INEQUALITIES FOR MULTIPLE INTEGRALS 5 Consequently, we get the identity 54) d c p x, s) q y, t) f s,t s, t) dtds d c) b ) f x, y) d c) b ) for ll x, y), b c, d d c f x, t) dt + Using the identity 54) we obtin, d 55) f s, t) dtds b ) Observe tht nd, similrly, c + d c) d c f s,t p x, s) ds d c d c d c f x, y) ds f s, t) dtds f x, t) dt f x, y) ds d c) b ) f x, y) p x, s) q y, t) f s,t s, t) dtds x d c p x, s) q y, t) dtds s ) ds + x ) + b x) x b s) ds 4 b ) + x + b ) q y, t) dt 4 d c) + y c + d ) Finlly, using 55), we hve the desired inequlity 59) Corollry 59 Under the bove ssumptions, we hve the inequlity: d d ) + b 56) f s, t) dtds b ) f c c, t dt + d c) f s, c + d ) + b ds d c) b ) f, c + d ) 6 b ) d c) f s,t Remrk 5 The constnts 4 from the first nd the second brcket re optiml in the sense tht not both of them cn be less thn 4

261 53 NS Brnett, P Cerone nd SS Drgomir Indeed, if we hd ssumed tht there exists c, c 0, 4) so tht d d 57) f s, t) dtds b ) f x, t) dt + d c) f s, y) ds c c d c) b ) f x, y) c b ) + x + b ) c d c) + y c + d ) f s,t for ll f s in Theorem 58 nd x, y), b c, d, then we would hve hd, for f s, t) st nd x, y c, d b ) d c ) f s, t) dtds, 4 c d f x, t) dt d c, c nd f s,t f s, y) ds c b, By 57), nd b ) d c ) b ) d c d c) c b + d c) b ) c 4 b ) c + ) d c) c + ) 4 4 giving ie 58) b ) d c) 4 b ) d c) c + ) c + ) c + ) c + ) 4 4 As we hve ssumed tht c, c 0, 4), we get c + 4 <, c + 4 < nd then c + 4) c + 4) < 4 which contrdicts the inequlity 58), nd estblishes the remrk 5 Remrk 5 If we ssume tht f s, t) h s) h t), h :, b R nd suppose tht h <, then from 59) we get for x y) h s) ds h s) ds h x) b ) h s) ds h x) b ) h s) ds + b ) h x) 4 b ) + x + b ) h

262 5 OSTROWSKI TYPE INEQUALITIES FOR MULTIPLE INTEGRALS 54 ie b h s) ds h x) b ) 4 b ) + x + b ) h which is clerly equivlent to Ostrowski s inequlity Consequently 59) cn be lso regrded s generliztion, for double integrls, of the clssicl result due to Ostrowski 5 Applictions for Cubture Formule Let us now consider the rbitrry division I n : x < x < < x n < x n b nd J m : c y 0 < y < < y m < y m b nd ξ i x i, x i+ i 0,, n ), η j y j, y j+ j 0,, m ) be intermedite points Consider the sum 59) n C f, I n, J m, ξ, η) : n + m j0 m j0 yj+ h i f ξ i, t) dt y j xi+ l j f n ) s, η j ds x i m j0 h i l j f ξ i, η j ) for which we ssume tht the involved integrls cn more esily be computed thn the originl double integrl nd D : d c f s, t) dsdt, h i : x i+ x i i 0,, n ), l j : y j+ y j j 0,, m ) With this ssumption, we cn stte the following cubture formul: Theorem 50 Let f :, b c, d R be s in Theorem 58 nd I n, J m, ξ nd η be s bove Then we hve the cubture formul: 50) d c f s, t) dtds C f, I n, J m, ξ, η) + R f, I n, J m, ξ, η) where the reminder term R f, I n, J m, ξ, η) stisfies the estimtion: 5) R f, I n, J m, ξ, η) f 4 n m s,t j0 f 4 h i + n m s,t h i j0 l j ξ i x ) i + x i+ 4 l j + η j y ) j + y j+

263 55 NS Brnett, P Cerone nd SS Drgomir Proof Apply Theorem 58 on the intervl x i, x i+ y j, y j+ i 0,, n ; j 0,, m ) to get: xi+ yj+ yj+ f s, t) dtds h i f ξ i, t) dt x i y j y j xi+ + l j f ) s, η j ds hi l j f ) ξ i, η j x i 4 h i + ξ i x ) i + x i+ 4 l j + η j y ) j + y j+ f s,t for ll i 0,, n ; j 0,, m Summing over i from 0 to n nd over j from 0 to m nd using the generlized tringle inequlity we deduce the first inequlity in 5) For the second prt we observe tht ξ i x i + x i+ h i for ll i, j s bove nd η j y j + y j+ l j Remrk 53 As nd where nd n n h i ν h) h i b ) ν h) m j0 m lj µ l) j0 l j d c) µ l) ν h) mx {h i : i 0,, n } µ l) mx {l j : j 0,, m }, the right membership of 5) cn be bounded by f s,t 4 b ) d c) ν h) µ l), which is of order precision Define the sum, C M f, I n, J m ) n m yj+ ) xi + x i+ : h i f, t dt + n j0 m j0 y j xi + x i+ h i l j f, y ) j + y j+, n m j0 xi+ l j f then we hve the best cubture formul possible from 50) x i s, y ) j + y j+ ds

264 5 OSTROWSKI TYPE INEQUALITIES FOR MULTIPLE INTEGRALS 56 Corollry 5 Under the bove ssumptions we hve 5) d c f s, t) dtds C M f, I n, J m ) + R f, I n, J m ) when the reminder R f, I n, J m ) stisfies the estimtion: f n m s,t R f, I n, J m ) h i lj 6 53 Some Inequlities in Terms of p Norm The following inequlity of Ostrowski s type for mppings of two vribles holds 3: Theorem 5 Let f :, b c, d R be continuous mpping on, b c, d, f x,y f x y exists on, b c, d nd is in L p, b c, d), ie, d ) f s,t p : p p f x, y) x y dxdy <, p > c then we hve the inequlity: d d 53) f s, t) dtds b ) f x, t) dt + d c) f s, y) ds c c d c) b ) f x, y) x ) q+ + b x) q+ q y c) q+ + d y) q+ q f s,t q + q + p for ll x, y), b c, d, where p + q j0 nd Proof If we consider the kernels: p :, b R, q : c, d R given by: s if s, x p x, s) : s b if s x, b t c if t c, y q y, t) : t d if s y, d then we hve the identity, see lso 54)) 54) d c p x, s) q y, t) f s,t s, t) dsdt d c) b ) f x, y) d c) b ) for ll x, y), b c, d d c f x, t) dt + d c f s, y) ds f s, t) dsdt

265 57 NS Brnett, P Cerone nd SS Drgomir From this we cn obtin the following inequlity:- d d f s, t) dsdt b ) f x, t) dt + d c) c c d c) b ) f x, y) d c p x, s) q y, t) f s,t s, t) dsdt Using Hölder s integrl inequlity for double integrls, we further hve, f s, y) ds d c d c p x, s) q y, t) f s,t s, t) dtds p x, s) q y, t) q dtds ) b q d p x, s) q ds x ) q+ + b x) q+ q + c ) q q y, t) q dt d c ) q f f s,t s, t) p dtds s,t p q y c) q+ + d y) q+ q + ) p q f s,t p nd the theorem is proved Corollry 53 Under the bove ssumptions, we hve the inequlity: d d ) + b 55) f s, t) dtds b ) f c c, t dt + d c) f s, c + d ) + b ds d c) b ) f, c + d ) b )+ q d c) + q 4q + ) q f s,t p Remrk 54 Consider the mpping g : α, β R, gt) t α) m + β t) m, m ) Tking into ccount the properties ) α + β inf gt) g β α)m t α,β m nd sup gt) gα) gβ) β α) m, t α,β 55) is seen to be the best inequlity tht cn be obtined from 53)

266 5 OSTROWSKI TYPE INEQUALITIES FOR MULTIPLE INTEGRALS 58 Remrk 55 Now, if we ssume tht f s, t) h s) h t), h :, b R is continuous on, b nd suppose tht h p <, then from 53) we get for x y) h s) ds h s) ds h x) b ) h s) ds h x) b ) h s) ds + b ) h x) x ) q+ + b x) q+ q h p q +, ie, b x ) h s) ds h x) b ) q+ + b x) q+ q h p q + which is clerly equivlent to Ostrowski s inequlity for p norms obtined in 6 54 Applictions For Cubture Formule Let us consider the rbitrry division I n : x 0 < x < < x n < x n b nd J m : c y 0 < y < < y m < y m b nd ξ i x i, x i+ i 0,, n ), η j y j, y j+ j 0,, m ) be intermedite points Consider the sum n C f, I n, J m, ξ, η) : n + m j0 m j0 yj+ h i f ξ i, t) dt y j xi+ l j f n ) s, η j ds x i m j0 h i l j f ξ i, η j ) for which we ssume tht the involved integrls cn more esily be computed thn the originl double integrl nd D : d c f s, t) dsdt, h i : x i+ x i i 0,, n ), l j : y j+ y j j 0,, m ) We cn stte the following cubture formul: Theorem 54 Let f :, b c, d R be s in Theorem 53 nd I n, J m, ξ nd η be s bove Then we hve, d c f s, t) dtds C f, I n, J m, ξ, η) + R f, I n, J m, ξ, η),

267 59 NS Brnett, P Cerone nd SS Drgomir where the reminder term R f, I n, J m, ξ, η) stisfies the inequlity estimtion), 56) R f, I n, J m, ξ, η) n f x i+ ξ i ) q+ + ξ i x i ) q+ s,t p q + m ) q+ ) q+ yj+ η j + ηj y j q + j0 f n s,t p h + m q q + ) i l + q j q j0 ) q ) q for ll ξ nd η s bove Proof Apply Theorem 5 to the intervl x i, x i+ y j, y j+ i 0,, n ; j 0,, m ) to get: xi+ yj+ f s, t) dtds x i y j yj+ xi+ h i f ξ i, t) dt + l j f ) s, η j ds hi l j f ) ξ i, η j y j x i ) ) x i+ ξ i ) q+ + ξ i x i ) q+ q+ ) q+ yj+ η j + ηj y j q + q + ) q xi+ x i yj+ y j f s, t) p dtds ) p, for ll i 0,, n ; j 0,, m

268 5 OSTROWSKI TYPE INEQUALITIES FOR MULTIPLE INTEGRALS 60 Summing over i from 0 to n nd over j from 0 to m nd using the generlized tringle inequlity nd Hölder s discrete inequlity for double sums, we deduce R f, I n, J m, ξ, η) ) n m x i+ ξ i ) q+ + ξ i x i ) q+ j0 q + yj+ η j ) q+ + ηj y j ) q+ q + ) q n x i+ ξ i ) q+ + ξ i x i ) q+ q + m ) q+ ) q+ yj+ η j + ηj y j q + j0 n n m j0 xi+ yj+ x i y j ) q f s, t) p dtds ) x i+ ξ i ) q+ + ξ i x i ) q+ m j0 q + yj+ η j ) q+ + ηj y j ) q+ q + To prove the second prt, we observe tht nd ) q p ) q x i+ x i y j+ y j f p x i+ ξ i ) q+ + ξ i x i ) q+ x i+ x i ) q+ yj+ η j ) q+ + ηj y j ) q+ yj+ y j ) q+ for ll i, j s bove nd the intermedite points ξ i nd η j We omit the detils s,t f s, t) p dtds p Remrk 56 As nd where nd n m j0 h + q i l + q j n ν h) q h i b ) ν h) q µ l) q m j0 l j d c) µ l) q ν h) mx {h i : i 0,, n }, µ l) mx {l j : j 0,, m },

269 6 NS Brnett, P Cerone nd SS Drgomir the right hnd side of 56) cn be bounded by Defining the sum, q + ) q f s,t n C M f, I n, J m ) : p b ) d c) ν h) µ l) q n + n m j0 m j0 m j0 h i l j y j+ y j x i+ x i ) xi + x i+ f, t dt f s, y ) j + y j+ ds xi + x i+ h i l j f we hve the best cubture formul possible from Theorem 54 Corollry 55 Under the bove ssumptions we hve, d c, y ) j + y j+, f s, t) dsdt C M f, I n, J m ) + R f, I n, J m ), where the reminder R f, I n, J m ) stisfies the inequlity estimtion), R f, I n, J m ) 4q + ) q f n s,t p h + q i m j0 l + q j 55 Some Inequlities in Terms of Norm The following result of Ostrowski s type lso holds Theorem 56 Let f :, b c, d R be continuous mpping on, b c, d, f x,y f x y exists on, b c, d nd is in L, b c, d), ie, d f s,t : f x, y) x y dxdy <, then we hve the inequlity, d 57) f s, t) dtds b ) c + d c) + x +b b for ll x, y), b c, d c + d c y c+d d c f x, t) dt f s, y) ds d c) b ) f x, y) b ) d c) f s,t

270 5 OSTROWSKI TYPE INEQUALITIES FOR MULTIPLE INTEGRALS 6 Proof As in the proof of Theorem 5, we use the inequlity d d 58) f s, t) dtds b ) f x, t) dt c + d c) d However, it is esy to see tht d c sup s,b c f s, y) ds d c) b ) f x, y) p x, s) q y, t) f s,t s, t) dtds p x, s) q y, t) f s,t s, t) dtds p x, s) sup q y, t) f s,t t c,d mx {x, b x} mx {d y, y c} f s,t b ) + x + b d c) + y c + d f s,t, which, vi 58), proves the desired inequlity 57) c The best inequlity we cn get from Theorem 56 is embodied in the following corollry Corollry 57 With the bove ssumptions, we hve, d d ) + b 59) f s, t) dtds b ) f c c, t dt + d c) f s, c + d ) + b ds d c) b ) f, c + d ) 4 b ) d c) f s,t Remrk 57 If we ssume tht f s, t) h s) h t), h :, b R is continuous on, b nd suppose tht h L, b, then from 57) we get for x y) h s) ds h s) ds h x) b ) h s) ds h x) b ) h s) ds + b ) h x) b ) + x + b h ie, b h s) ds h x) b ) b ) + x + b h which is clerly equivlent to Ostrowski s inequlity for norms obtined in 7,

271 63 NS Brnett, P Cerone nd SS Drgomir Applictions for cubture formule cn be provided in similr fshion s bove, but we omit the detils 53 Other Ostrowski Type Inequlities 53 Some Identities The following theorem holds 5 Theorem 58 Let f :, b c, d R be such tht the prtil derivtives ft,s) ft,s) s, ft,s) t s t, exist nd re continuous on, b c, d Then for ll x, y), b c, d, we hve the representtion 530) f x, y) b ) d c) + b ) d c) + b ) d c) + b ) d c) d c d c d c d c f t, s) dsdt p x, t) q y, s) where p :, b R, q : c, d R nd re given by t if t, x 53) p x, t) : t b if t x, b nd s c if s c, y 53) q y, s) : s d if s y, d f t, s) dsdt t f t, s) dsdt s p x, t) q y, s) f t, s) dsdt, t s Proof We use the following identity, which cn be esily proved using integrtion by prts 533) g u) β g z) dz + β k u, z) g z) dz, β α α β α α where k : α, β R is given by z α if z α, u k u, z) : z β if z u, β nd g is bsolutely continuous on α, β, Indeed, we hve u z α) g z) dz u α) g u) nd α β u u α g z) dz β z β) g z) dz β u) g u) g z) dz u

272 5 OSTROWSKI TYPE INEQUALITIES FOR MULTIPLE INTEGRALS 64 which produces, by summtion, the desired identity 533) We cn write the identity 533) for the prtil mp f, y), y c, d to obtin 534) f x, y) b for ll x, y), b c, d Similrly, 535) f t, y) d c for ll t, y), b c, d d c f t, y) dt + b f t, s) ds + d c d c p x, t) q y, s) The sme formul 533) pplied for the prtil derivtive f,y) t 536) f t, y) t for ll t, y), b c, d d d c c f t, s) ds + t d c d c f t, y) dt t f t, s) ds, s gives, q y, s) f t, s) ds t s Substituting 535) nd 536) in 534), nd using Fubini s theorem, we hve b d f x, y) f t, s) ds + d f t, s) q y, s) ds dt b d c c d c c s + b d f t, s) p x, t) ds b d c c t + d q y, s) f t, s) ds dt d c c t s b d d f t, s) f t, s) dsdt + q y, s) dsdt b ) d c) s + d c p x, t) nd the identity 530) is estblished c f t, s) dsdt + t d c c p x, t) q y, s) f t, s) dsdt t s, A prticulr cse which is of interest is embodied in the following corollry Corollry 59 Let f be s in Theorem 58, then we hve the identity + b f, c + d ) 537) b d d f t, s) f t, s) dsdt + p 0 t) dsdt b ) d c) c c t d f t, s) b d + q 0 s) dsdt + p 0 t) q 0 s) f t, s) dsdt, s t s c c

273 65 NS Brnett, P Cerone nd SS Drgomir where p 0 :, b R, q 0 : c, d R re given by p 0 t) : nd q 0 s) : t if t, +b t b if t +b, b, s c if s c, c+d s d if s c+d, d The following corollry, which provides trpezoid type identity, is lso of interest Corollry 50 Let f be s in Theorem 58 Then we hve the identity 538) f, c) + f, d) + f b, c) + f b, d) 4 b d d f t, s) dsdt + t + b ) f t, s) dsdt b ) d c) c c t d + s c + d ) f t, s) dsdt c s d + t + b ) s c + d ) f t, s) dsdt t s c Proof Letting x, y), c),, d), b, c) nd b, d) in 530), we obtin successively, f, c) b b ) d c) + d c s d) d c f t, s) dsdt + f t, s) dsdt + s d d c c t b) f t, s) dsdt t t b) s d) f t, s) dsdt s t, f, d) b b ) d c) + d c s c) d c f t, s) dsdt + f t, s) dsdt + s d d c c t b) f t, s) dsdt t t b) s c) f t, s) dsdt s t, f b, c) b b ) d c) + d c s d) d c f t, s) dsdt + f t, s) dsdt + s d d c c t ) f t, s) dsdt t t ) s d) f t, s) dsdt s t,

274 5 OSTROWSKI TYPE INEQUALITIES FOR MULTIPLE INTEGRALS 66 nd f b, d) b b ) d c) + d c s c) d c f t, s) dsdt + f t, s) dsdt + s d d c c t ) f t, s) dsdt t t ) s c) f t, s) dsdt s t After summing over the bove equlities, dividing by 4 nd some simple computtion, we rrive t the desired identity 538) 53 Some Bounds We cn stte the following inequlity of the Ostrowski type which holds for mppings of two independent vribles 5 Theorem 5 Let f :, b c, d R be mpping s in Theorem 58 Then we hve the inequlity: 539) f x, y) b ) d c) d M x) + M y) + M 3 x, y), c f t, s) dsdt where M x) 4 b ) + ) x +b b b x) q + +x ) q + q + b ) d c) p q b ) + x +b b ) d c) f t, f t, p f t, if if if f t, s) t f t, s) t L, b c, d) ; L p, b c, d), p >, p + q ; f t, s) L, b c, d) t M y) 4 d c) + ) y c+d d c d y) q + +y c) q + q + b ) p d c) q d c) + y c+d b ) d c) f s, f s, p f s, if if if f t, s) s L, b c, d) ; f t, s) L p, b c, d) ; s p >, p + q ; f t, s) L, b c, d) ; s

275 67 NS Brnett, P Cerone nd SS Drgomir nd M 3 x, y) 4 b ) + ) x +b 4 d c) + ) y c+d f b ) d c) t s, f t, s) if L, b c, d) ; s t b x) q 3 + +x ) q 3 + q 3 d y) q 3 + +y c) q 3 + q 3 q 3+ q 3+ f b ) d c) t s, p3 f t, s) if L p3, b c, d), p 3 >, s t p 3 + q 3 ; b ) + x +b d c) + y c+d f b ) d c) t s, f t, s) if L, b c, d) ; s t for ll x, y), b c, d, where p p ) re the usul p norms on, b c, d Proof Using the identity 530), we cn stte tht f x, y) b d 540) f t, s) dsdt b ) d c) c b d f t, s) p x, t) dsdt b ) d c) t We hve, 54) d c d f t, s) + q y, s) dsdt + p x, t) q y, s) f t, s) dsdt c s c t s b d p x, t) f t, s) b ) d c) c t dsdt d + q y, s) f t, s) b d s dsdt + p x, t) q y, s) f t, s) t s dsdt c d c f t f t f t p x, t) p f t, s) t dsdt c d c p x, t) dsdt, if ft,s) t L, b c, d) ; ) d c p x, q dsdt, if ft,s) t) q t L p, b c, d), sup t,b p x, t), if p >, p + q ; ft,s) t L, b c, d)

276 5 OSTROWSKI TYPE INEQUALITIES FOR MULTIPLE INTEGRALS 68 nd s d d c p x, t) dsdt c ) b x p x, t) dt ds d c) p x, t) dt + x d c) t ) dt + d c) 4 b ) + x b t) dt x + b ), x p x, t) dt d c p x, t) q dsdt q d ) b p x, t) q dt ds c d c) q d c) q d c) q q x p x, t) q dt + x t ) q dt + sup p x, t) mx {x, b x} b t,b nd then, by 54), we obtin 54) d c p x, t) x b x) q+ + x ) q+ f t, s) t dsdt q + x p x, t) q dt b t) q dt + x + b, q q q 543) d c) 4 b ) + ) x +b f t, if ft,s) t L, b c, d) ; d c) q b x) q + +x ) q + q q + b ) + x +b f t f, p t if if ft,s) t L p, b c, d), p >, p + q ; ft,s) t L, b c, d) In similr fshion, we cn stte, 544) d c q y, s) f t, s) s dsdt

277 69 NS Brnett, P Cerone nd SS Drgomir 545) b ) 4 d c) + ) y c+d f s, if ft,s) s L, b c, d) ; b ) q d y) q + +y c) q + q q + d c) + y c+d f s f, p s if if ft,s) s L p, b c, d), p >, p + q ; ft,s) s L, b c, d) In ddition, we hve 546) d c p x, t) q y, s) f t, s) t s dsdt 547) ft,s) t s if p x, t) dt d q y, s) ds, ft,s) s t L, b c, d) ; ft,s) t s if p3 p x, dt t) q3 c ) q 3 ) d c q y, q 3 ds, s) q3 ft,s) s t L p3, b c, d), p 3 >, p 3 + q 3 ; sup t,b p x, t) sup s c,d q y, s) ft,s) t s if ft,s) s t L, b c, d) 4 b ) + ) x +b 4 d c) + ) y c+d f t s, if ft,s) s t L, b c, d) ; b x) q 3 + +x ) q 3 + q 3 d y) q 3 + +y c) q 3 + q 3+ q 3+ q 3 f t s if ft,s) s t L p3, b c, d) ; p 3 >, p 3 + q 3 ; b ) + x +b d c) + y c+d f t s,, p3 if nd the theorem is proved ft,s) s t L, b c, d) The following corollry holds by tking x +b, y c+d

278 5 OSTROWSKI TYPE INEQUALITIES FOR MULTIPLE INTEGRALS 70 Corollry 5 With the ssumptions in Theorem 58, we hve the inequlity + b f, c + d ) b d 548) f t, s) dsdt b ) d c) where nd M : M : M 3 : M + M + M 3, 4 b ) f t, b ) q q + ) q d c) p d c) f t, d c) f s, d c) q q + ) q b ) p b ) f s, b ) d c) f t s, b ) f t s, q 3 d c) q 3 q 3 + ) q 3 f t, p f s p f t s, p3 if if if if if if if if if c f t L, b c, d) f t L p, b c, d) p >, p + q ; f t L, b c, d) f t L, b c, d) f t L p, b c, d) p >, p + q ; f t L, b c, d) f t s L, b c, d) ; f t s L p 3, b c, d) ; p 3 >, p 3 + q 3, f t s L, b c, d) Using the inequlity 538) in Corollry 50 nd similr rgument to the one used in Theorem 5, we hve the following trpezoid type inequlity: Corollry 53 With the ssumption in Theorem 58, we hve the inequlity f, c) + f, d) + f b, c) + f b, d) b d 549) f t, s) dsdt 4 b ) d c) M + M + M 3, c where M i i,, 3) re s given bove

279 7 NS Brnett, P Cerone nd SS Drgomir 533 Applictions for Cubture Formule Consider the rbitrry divisions I n : x 0 < x < < x n < x n b nd J m : c y 0 < y < < y m < y m d, where ξ i x i, x i+ i 0,, n ), η j y j, y j+ j 0,, m ) re intermedite points Consider further the Riemnn sum: n 550) R f, I n, J m, ξ, η) m j0 h i l j f ξ i, η j ), where h i : x i+ x i, l j : y j+ y j, i 0,, n, j 0,, m Using Theorem 5, we cn stte twenty-seven different inequlities bounding the quntity 55) f x, y) b ) d c) d c f t, s) dsdt, x, y), b c, d Consider one, nmely, when ll the prtil derivtives f We then hve:-, t, f s, f t s re bounded 55) f x, y) b ) d c) b + d c 4 b ) + 4 d c) + + b ) d c) 4 d c) + d x + b f t, s) dsdt c ) f t ) f s x + b ) y c + d 4 b ) + y c + d ) f t s for ll x, y, b c, d Using this inequlity, we cn stte the following theorem 5 Theorem 54 Let f :, b c, d R be s in Theorem 58, then we hve 553) d c f t, s) dsdt R f, I n, J m, ξ, η) + W f, I n, J m, ξ, η),

280 5 OSTROWSKI TYPE INEQUALITIES FOR MULTIPLE INTEGRALS 7 where R f, I n, J m, ξ, η) is the Riemnn sum defined by 550) nd the reminder, through the pproximtion W f, I n, J m, ξ, η), stisfies, 554) W f, I n, J m, ξ, η) n d c) f t + b ) f s + f t s m j0 m j0 n 4 l j + d c) f t 4 h i + 4 h i + 4 l j + η j y j + y j+ ξ i x i + x i+ ) η j y j + y j+ ) ξ i x i + x i+ ) n h i + b ) f s ) m lj j0 + n f m 4 t s h i lj j0 ν d c) b ) h) f t + ν l) f s + f t s ν h) ν l) for ll ξ, η intermedite points, where ν h) : mx {h i, i 0,, n } nd ν l) : mx {l j, j 0,, m } Proof Apply 55) in the intervls x i, x i+ y j, y j+ to obtin xi+ yj+ f t, s) dsdt h i l j f ) ξ i, η j x i y j 4 h i + ξ i x ) i + x i+ f l j t + 4 l j + η j y ) j + y j+ f h i s + 4 h i + ξ i x ) i + x i+ 4 l j + η j y ) j + y j+ f t s for ll i 0,, n, j 0,, m Summing over i from 0 to n nd over j from 0 to m, we get the desired estimtion 554) Consider the mid-point formul: n 555) M f, I n, J m ) : m j0 xi + x i+ h i l j f, y ) j + y j+

281 73 NS Brnett, P Cerone nd SS Drgomir The following corollry contins the best qudrture formul we cn obtin from 554) Corollry 55 Let f be s in Theorem 58, then we hve: 556) d c f t, s) dsdt M f, I n, J m ) + L f, I n, J m ), where M f, I n, J m ) is the midpoint formul given by 555) nd the reminder L f, I n, J m ) stisfies the estimte 557) L f, I n, J m ) 4 d c) f t + 6 f t s n h i + 4 b ) f s n m h i lj j0 m lj j0 : M f, I n, J m ) ν 4 d c) b ) h) f t + ν l) f s + 4 ν h) ν l) f t s : M f, I n, J m ) We cn lso consider the trpezoid formul 558) T f, I n, J m ) n : m j0 h i l j f x i, y j ) + f x i, y j+ ) + f x i+, y j ) + f x i+, y j+ ) 4 Using Corollry 53 nd similr rgument to the one used in the proof of Theorem 54, we cn stte the following, Corollry 56 Let f be s in Theorem 58, then we hve 559) d c f t, s) dsdt T f, I n, J m ) + P f, I n, J m ), where T f, I n, J m ) is the trpezoid formul obtined by 558) nd the reminder P f, I n, J m ) stisfies the estimte 560) P f, I n, J m ) M f, I n, J m ) M f, I n, J m ), where M nd M re s defined bove 54 Ostrowski s Inequlity for Hölder Type Functions 54 The Unweighted Cse We strt with the following Ostrowski type inequlity for mppings of the r Hölder type see 4)

282 5 OSTROWSKI TYPE INEQUALITIES FOR MULTIPLE INTEGRALS 74 Theorem 57 Assume tht the mpping f :, b n, b n R stisfies the following r Hölder type condition: H) n f x) f ȳ) L i x i y i ri L i 0, i,, n) i for ll x x,, x n ), ȳ y,, y n ) ā, b :, b n, b n, where r i 0,, i,, n We hve then the Ostrowski type inequlity: 56) f x) n i L i r i + n b i i ) xi i i b i i n L i b i i ) ri, r i + i b ā ) ri+ f t) d t ) ri+ bi x i + b i i ) ri b i i for ll x ā, b, where b ā f t) d t f t,, t n ) dt n dt Proof By H) we hve n f x) f t) L i x i t i ri i for ll x, t ā, b Integrting over t on ā, b nd using the modulus properties we get 56) f x) b d t b ā ā f t) d t b ā i f x) f t) d t n L i n n x i t i ri dt n dt As n dt n dt n n b i i ) i

283 75 NS Brnett, P Cerone nd SS Drgomir nd n x i t i ri dt n dt n n i b j j ) x i t i ri dt i j j i i n b i x i ) ri+ + x i i ) ri+ b j j ) r i + j j i n b j j ) r i + j then, dividing 56) by n j Using the elementry inequlity bi x i b i i ) ri+ ) ri+ xi i + b i i ) ri, b i i b j j ) we get the first prt of 56) y α) p+ + β y) p+ β α) p+ for ll α y β nd p > 0, we obtin, ) ri+ bi x i xi i + b i i b i i nd the lst prt of 56) is lso proved ) ri+, i,, n Some prticulr cses re interesting Corollry 58 Under the bove ssumptions, we hve the mid-point inequlity: + b f,, ) n + b n b n 563) n f t,, t n ) dt n dt b i i ) n n i L i b i i ) ri ri r i + ), i which is the best inequlity we cn get from 56) Proof Note tht the mpping h p : α, β R, h p y) y α) p+ + β y) p+ p > 0) hs its infimum t y 0 α+β nd inf h β α)p+ p y) y α,β p Consequently, the best inequlity we cn get from 56) is the one for which x i i+b i giving the desired inequlity 563) The following trpezoid type inequltiy lso holds

284 5 OSTROWSKI TYPE INEQUALITIES FOR MULTIPLE INTEGRALS 76 Corollry 59 Under the bove ssumptions, we hve: f,, n ) + f b,, b n ) b n 564) n f t,, t n ) dt n dt b i i ) n n L i b i i ) ri r i + i i Proof Put in 56) x ā nd then x b, dd the obtined inequlities nd use the tringle inequlity to get 564) An importnt prticulr cse is one for which the mpping f is Lipschitzin, ie, 565) f x,, x n ) f y,, y n ) for ll x, ȳ ā, b Thus, we hve following corollry n L i x i y i Corollry 530 Let f be Lipschitzin mpping with the constnts L i Then we hve b ) 566) n f x) n f t) d t L i b i i ) ā 4 + xi i+bi b i i ) b i i i i for ll x ā, b i The constnt 4, in ll the brckets, is the best possible Proof Choose r i i,, n) in 56) to get b n f x,, x n ) n f t,, t n ) dt n dt b i i ) n i n xi ) ) i bi x i L i + b i i ) b i i b i i i A simple computtion shows tht xi ) ) i bi x i + 4 b i i b i + xi i+bi i b i i giving the desired inequlity 566) ), i,, n

285 77 NS Brnett, P Cerone nd SS Drgomir To prove the shrpness of the constnts 4, ssume tht the inequlity 566) holds for some positive constnts c i > 0, ie, 567) b ) n f x) n f t) d t xi L i c i+bi i + b i i ), b i i ) ā b i i i i for ll x ā, b Choose f x,, x n ) x i i,, n) Then, by 567), we get x i i + b i c i + for ll x i i, b i Put x i i, to get b i i xi i+bi b i i ) ) c i + ) b i i ) 4 b i i ) from which we deduce c i 4, nd the shrpness of 4 is proved Corollry 53 If f is s in Corollry 530, then we get ) the mid-point formul + b f,, ) n + b n 568) n b i i ) 4 n L i b i i ), i i f t,, t n ) dt n dt n n b) the trpezoid formul f,, n ) + f b,, b n ) n L i b i i ) i n b i i ) i f t,, t n ) dt n dt n Remrk 58 In prcticl pplictions, we ssume tht the mpping f : ā, b R hs the prtil derivtives f x i nd f x i bounded on ā, b, ie f x i : sup f x,, x n ) x i < x ā, b) n

286 5 OSTROWSKI TYPE INEQUALITIES FOR MULTIPLE INTEGRALS 78 With this ssumption, we hve the Ostrowski type inequlity see Introduction) b f x) n f t) d t b i i ) ā i n ) f x i 4 + xi i+bi b i i ) b i i i The constnts, 4, re shrp 54 The Weighted Cse The following generliztion of Theorem 57 holds see 4) Theorem 53 Let f, w : ā, b R be such tht f is of the r Hölder type with the constnts L i nd r i 0, i,, n) nd where w is integrble on ā, b, nonnegtive on this intervl nd b ā w x) d x : We then hve the inequlity, 569) f x) b w ȳ) dȳ for ll x ā, b ā b ā n n w x,, x n ) dx n dx > 0 w ȳ) f ȳ) dȳ n i Proof The proof is similr to tht of Theorem 57 b ā L x i y i ri w ȳ) dȳ i b ā w ȳ) dȳ As f is of the r Hölder type with the constnts L i nd r i i,, n), we cn write n f x) f ȳ) L i x i y i ri for ll x, ȳ ā, b Multiplying by w ȳ) 0 nd integrting over ȳ on ā, b, we get b n b 570) f x) f ȳ) w ȳ) dȳ L i x i y i ri w y i,, y n ) dy n dy ā On the other hnd, we hve 57) b ā b ā i i f x) f ȳ) w ȳ) dȳ f x) f ȳ)) w ȳ) dȳ f x) ā b ā w ȳ) dȳ b ā f ȳ) w ȳ) dȳ Combining 570) with 57) nd dividing by b w ȳ) dȳ > 0 we get the desired ā inequlity 569)

287 79 NS Brnett, P Cerone nd SS Drgomir Remrk 59 If we ssume tht the mpping f is Lipschitzin with constnts L i then we get, b f x) n b w ȳ) f ȳ) dȳ b w ȳ) dȳ ā ā L x i y i ri w ȳ) dȳ i, b ā i ā w ȳ) dȳ which generlizes Milovnović result from 975 see Introduction) The following corollries hold Corollry 533 With the ssumptions from Theorem 53 we hve: b f x) 57) w ȳ) f ȳ) dȳ b w ȳ) dȳ ā ā n bi i L i + x i i + b i for ll x ā, b Then, Proof We hve b ā i b x i y i ri w ȳ) dȳ sup x i y i ri ȳ ā, b ā n i mx { x i i, x i b i } mx {x i i, b i x i } b ā L x i y i ri w ȳ) dȳ i b w ȳ) dȳ ā w ȳ) dȳ b ā b ā w ȳ) dȳ w ȳ) dȳ bi i + x i i + b i b w ȳ) dȳ n i L i bi i nd by 569), we get the desired estimtion 57) Another type of inequlity estimtion) is s follows ā + x i i + b i Corollry 534 With the ssumptions from Theorem 53, we hve b f x) 573) w ȳ) fȳ)dȳ b w ȳ) dȳ ā ā n sup w ȳ) b b j j ) ā w ȳ) dȳ ȳ ā, b j n b i x i ) ri+ + x i i ) ri+ L i r i + ) b i i ) for ll x ā, b i

288 5 OSTROWSKI TYPE INEQUALITIES FOR MULTIPLE INTEGRALS 80 Proof We observe tht b ā x i y i ri w ȳ) dȳ sup w ȳ) ȳ ā, b n b j j ) Now, using 569) we get 573) j j i b ā i i x i y i ri dȳ x i y i ri dy i n b i x i ) ri+ + x i i ) ri+ b j j ) r i + j j i n b i x i ) ri+ + x i i ) ri+ b j j ) r i + ) b i i ) j Finlly, by Hölder s integrl inequlity we cn lso stte the following corollry: Corollry 535 With the ssumptions from Theorem 53, we hve b f x) 574) w ȳ) fȳ)dȳ b w ȳ) dȳ ā ā ) /q b ā w ȳ)q dȳ n n b b j j ) b p i x i ) pri+ + x i i ) pri+ L i w ȳ) dȳ pr ā j i i + ) b i i ) for ll x ā, b Proof Using Hölder s integrl inequlity for multiple integrls, we get b ) /q b b 575) x i y i ri w ȳ) dȳ w ȳ) q dȳ x i y i pri dȳ However, b ā ā x i y i pri dȳ nd then, by 575), we get b ā x i y i pri dȳ ā n b j j ) j j i i i x i y i ri dy i n b i x i ) pri+ + x i i ) pri+ b j j ) pr i + ) b i i ) j n b b j j ) p w ȳ) q dȳ j ā ) Using 569), we deduce the desired estimtion574) q b i x i ) pri+ + x i i ) pri+ ā pr i + ) b i i ) ) p p p

289 Bibliogrphy NS BARNETT nd SS DRAGOMIR, An Ostrowski type inequlity for double integrls nd pplictions for cubture formule, Soochow J Mth, in press) NS BARNETT nd SS DRAGOMIR, A note on bounds for the estimtion error vrince of continuous strem with sttionry vriogrm, J KSIAM, ) 998), SS DRAGOMIR, NS BARNETT nd P CERONE, An Ostrowski type inequlity for double integrls in terms of L p-norms nd pplictions in numericl integrtion, Anl Num Theor Approx Romni), in press) 4 SS DRAGOMIR, NS BARNETT nd P CERONE, An n dimensionl version of ostrowski s inequlity for mppings of the Hölder type, Kyungpook Mth J, 40) 000), SS DRAGOMIR, P CERONE, NS BARNETT nd J ROUMELIOTIS, An inequlity of the Ostrowski type for double integrls nd pplictions to cubture formule, Tmkng Oxford J Mth Sci, 6) 000), -6 6 SS DRAGOMIR nd S WANG, A New Inequlity of Ostrowski s type in L p norm, Indin J Mth, 403) 998), SS DRAGOMIR nd S WANG, A new inequlity of Ostrowski s type in L norms nd pplictions to some specil mens nd some numericl qudrture rules, Tmkng J Mth, 8 997), GV MILOVANIĆ, On some integrl inequlities, Univ Beogrd Publ Elek Fk, Ser Mt Fiz, No ), DS MITRINOVIĆ, JE PE CARIĆ nd AM FINK, Inequlities Involving Functions nd Their Integrls nd Derivtives, Kluwer Acdemic Publishers, Dordrecht, 994 8

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291 CHAPTER 6 Some Results for Double Integrls Bsed on n Ostrowski Type Inequlity by G HANNA Abstrct An Ostrowski type inequlity in two dimensions for double integrls on rectngle region is developed The resulting integrl inequlities re evluted for the clss of functions with bounded first derivtive They re employed to pproximte the double integrl by one dimensionl integrls nd function evlutions using different types of norms If the one-dimensionl integrls re not known, they themselves cn be pproximted by using suitble rule, to produce cubture rule consisting only of smpling points In ddition, some generlistions of n Ostrowski type inequlity in two dimensions for n - time differentible mppings re given The result is n integrl inequlity with bounded n - time derivtives This is employed to pproximte double integrls using one dimensionl integrls nd function evlutions t the boundry nd interior points 6 Introduction In this chpter three point cubture rules for two-dimensionl rectngulr regions re developed An priori error bound is obtined for functions whose first prtil derivtives exist nd re bounded The term three point is used to drw n nlogy with Newton-Cotes type rules where smpling occurs t the boundry nd interior points The rule presented here pproximtes two-dimensionl integrl vi ppliction of function evlutions nd one-dimensionl integrls t the boundry nd interior points And prmeteriztion, similr to tht of 5, is employed to distinguish rule type If the one-dimensionl integrls re not known, they themselves cn be pproximted to produce cubture rule consisting only of smpling points An dditionl three point rule, s in 5, my be subsequently used, or indeed ny other desired qudrture rule For exmple, the optiml rules in7, 0) As result the error bound will be lrger The method presented here is bsed on Ostrowski s integrl inequlity, nd s such is menble to the production of error bounds for vriety of norms In ddition 83

292 6 DOUBLE INTEGRAL INEQUALITIES 84 smoother nd product integrnds my lso be considered s hs been done for one-dimensionl integrls, see for exmple 5, 6, 9 6 The One Dimensionl Ostrowski Inequlity The clssicl Ostrowski integrl inequlity in one dimension stipultes bound between function evluted t n interior point x nd the verge of the function of over n intervl The Ostrowski inequlity hs been extended nd generlized in mny wys -usully by plcing higher demnds on the mpping f smoothness, monotonicity, etc) Here we focus on two such extensions In 5, where Cerone nd Drgomir presented three point inequlity nd showed tht the tightest bound is n verge of the mid-point nd trpezoidl rules In the pper 4, Brnett nd Drgomir developed two dimensionl version of the Ostrowski inequlity The current work combines the bove two results nd develops two dimensionl three point integrl inequlity for functions with bounded first derivtives An ppliction in the numericl integrtion of two-dimensionl integrl is investigted Also, some generlistions of n Ostrowski type inequlity in two dimensions for n time differentible mppings re given The result is n integrl inequlity with bounded n time derivtives This is employed to pproximte double integrls using one dimensionl integrls nd function evlutions t the boundry nd interior points The Chpter is rrnged in the following mnner In Section 63, n inequlity for double integrls is obtined in terms of first derivtives where t t f t, f t ) L, b, b Some Numericl results re computed in Section 64 An ppliction for the cubture formul is illustrted in Section 65 In Section 66, n inequlity is developed for mppings whose first derivtives t t f t, f t ) L p, b, b An ppliction is demonstrted through numericl results in Section 67 Section68 is reserved for results involving mppings whose first derivtives belong to -norm In Section 69 some generl identities in two dimention for n-time differentible mpping re given These re then pplied to produce some generliztions of Ostrowski s type inequlities using different types of norms in Section 60 These results re employed to produce pplictions to numericl integrtion s in Section 6 63 Mpping Whose First Derivtives Belong to L, b) Theorem 6 Let f : R R be differentible mpping on, b, b nd let f t,t f t t be bounded on, b ), b ) Tht is, f t,t : sup f t t < x,x ),b ),b ) Furthermore, let x i i, b i ) nd introduce the prmeteriztion α i, β i defined by 6) α i γ i ) i + γ i x i, β i γ i ) b i + γ i x i,

293 85 G Hnn where γ i 0,, for i, Then the following inequlity holds 6) Gx, t, x, t ) f t,t 4 + γ ) ) b + γ ) ) b ) + x + b ) ) + x + b ), given tht 63) Gx, t, x, t ) 64) f jk ) 65) C jk ) 66) I jk ) 3 k j 3 C k C j f jk 3 C j I j + C j I j ) j f, ) f x, ) f b, ) f, x ) f x, x ) f b, x ) f, b ) f x, b ) f b, b ) + f t, t ) dt dt, γ x ) γ x ) γ ) b ) γ ) b ) γ b x ) γ b ) ft, ) dt f, t ) dt ft, x ) dt fx, t ) dt ft, b ) dt fb, t ) dt, Proof Define the kernel 67) p x, t) { t α, t, x, t β, t x, b, where, s bove, α γ) + γx, nd β γ) b + γx Using 67) nd integrting by prts we obtin, fter some simplifiction, the identity 68) p x, t) F t) dt γ) b ) F x) + γ x ) F ) + b x) F b) F t) dt A two dimensionl identity cn be developed vi repeted ppliction of 68) To this end, we define the mpping { ti α 69) p i x i, t i ) i, i t i x i, for i, t i β i, x i < t i b i,

294 6 DOUBLE INTEGRAL INEQUALITIES 86 Substituting p for p nd ft, ) for F t) into 68) gives 60) p x, t ) f t dt γ ) b ) f x, t ) + γ x ) f, t ) + γ b x ) f b, t ) f t, t ) dt Employing 68) gin with p s the kernel, F t ) p x, t ) f t dt s the integrnd nd expnding with 60) produces, p x, t )F t ) dt p x, t ) p x, t ) f t t dt dt γ ) b ) F x ) + γ b x ) F b ) + x ) F ) F t ) dt γ ) γ ) b ) b ) f x, x ) + γ γ ) b ) b x ) f b, x ) + γ γ ) b ) x ) f, x ) + γ γ ) b ) b x ) f x, b ) + γ γ b x ) b x ) f b, b ) + γ γ b x ) x ) f, b ) + γ γ ) x ) b ) f x, ) + γ γ x ) b x ) f b, ) + γ γ x ) x ) f, ) γ ) b ) γ x ) f t, x ) dt γ b x ) f t, ) dt γ ) b ) γ b x ) f b, t ) dt γ x ) C k C j f jk C j I j + C j I j ) + k j nd this produces tht j f t, b ) dt f x, t ) dt f, t ) dt + ft, t ) dt dt f t, t ) dt dt 6) p x, t ) p x, t ) f t t dt dt Gx, t, x, t ) Assuming tht both first prtil derivtives of f re bounded, we cn simply write down the inequlity 6) f p x, t ) p x, t ) dt dt t t f t,t p x, t ) dt ) p x, t ) dt )

295 87 G Hnn Now, consider G x ) p x, t ) dt α x β t α ) dt + t α ) dt t β ) dt + α x α ) + x α ) + β x ) + b β ) 63) + γ ) ) b + x ) + b Similrly, with G x ) p x, t ) dt, we hve 64) G x) + γ ) b ) + x ) + b β t β ) dt Using 64), 65) nd 66) nd substituting 6), 63) nd 64) into 6) will produce the result 6) nd thus the theorem is proved The following result gives n Ostrowski type inequlity for double integrls It involves double nd single integrls together with function evlution t n interior point Corollry 6 With the conditions s in Theorem 6, then 65) b ) b ) f x, x ) b ) f t, x ) dt b ) f x, t ) dt + f t, t ) dt dt b ) f t,t + x ) b ) + b + x ) + b Proof Plce γ γ 0 into eqution 6) Thus, the erlier results of 4 nd 8, p 468 re reproduced s specil cse of Theorem 6 We note tht unlike 4, the proof for Theorem 6 cn be esily extended to more thn two dimensions Different vlues of the prmeters γ, γ, x nd x give rise to Newton-Cotes type inequlities for functions with bounded derivtives For exmple γ γ 0, x +b nd x +b produces the two dimensionl mid-point inequlity; γ γ two dimensionl trpezoid-like inequlity nd γ γ 3 two dimensionl Simpson s like inequlity

296 6 DOUBLE INTEGRAL INEQUALITIES 88 From Theorem 3 it is simple mtter to show tht the tightest bound is obtined when γ γ nd x nd x re t their mid-points Tht is for the verge of the mid-point nd trpezoid inequlities Remrk 6 Let f t, t ) g t ) g t ) where g :, b R If g is differentible nd stisfies the condition tht g <, then, for x x x nd γ γ γ, we obtin result from Theorem 6 which my be fctored to recover the three point rule developed in 5, nmely 66) gt) dt γ x )g) + b x)gb) ) γ)b )gx) g + γ ) ) ) b + x + b ) ) In generl, cubture formule re written only in terms of function evlutions, but Theorem 6 pproximtes double integrl in terms of single integrls nd function evlutions Therefore we write down the following corollry which elimintes the one dimensionl integrls by pproximting them using the 3-point rule in eqution 66) The resulting inequlity hs corser bound thn eqution 6) Corollry 63 Let f be given s in Theorem 6 Then 67) f t, t ) dt dt f t,t 3 k j 3 C k C j f jk + γ 4 ) ) + γ ) ) b ) + x ) b ) + b + x ) + b + + γ ) ) ) b + x ) + b {γ x ) f t, + γ ) b ) f t,x + γ b x ) } f t,b + + γ ) ) ) b + x ) + b {γ x ) f,t + γ ) b ) f x,t + γ b x ) } f b,t Proof Approximting ech single integrl in 6) by 66) nd pplying the tringle inequlity produces the desired result

297 89 G Hnn Remrk 6 If γ γ 0 nd x i i+bi, then 67) becomes + b 68) f t, t ) dt dt b ) b ) f, ) + b f t f,t b ) b ) t, +b + b ) b ) 6 4 f +b,t + b ) b ) 4 64 Numericl Results In this section the inequlities developed in Section 63 re used to pproximte the double integrl 69) 0 0 e xy dxdy This integrnd ws chosen becuse integrting once in ech direction is trivil Nmely, 0 e xy dx y+e y y nd 0 e xy dy x+e x x, but the double integrl is not In Tble 6, results re shown for the pproximtion to 69) using the rule nd bound of 6) The numericl error is much smller thn the theoreticl one nd is smllest when Simpson s rule is pplied γ γ 3 ) The optiml theoreticl bound is ttined when γ γ It should be noted tht γ γ 0 pproximtes 69) with the mid-point rule nd employs one function evlution t the midpoint of the region) nd two one-dimensionl integrls long the bi-sectors) The trpezoidl rule uses four smple points the boundry corners) nd four one-dimensionl integrls long the boundry) All other vlues, tht is γ, γ 0, ), produces rule tht is liner combintion of the bove nd results in the use of nine smple points nd six one-dimensionl integrls γ γ Numericl Error Theoreticl Error ) 63-) ) 9-) ) 6-) 65-3) 63-) Tble 6 The numericl nd theoreticl errors in computing 69) using 6) with x x 05 nd vrious vlues of γ, γ To pproximte 69) only in terms of function evlutions we use eqution 67) The results re presented in Tble 6 Since 67) is n pproximtion of 6), the results re qulittively similr nd quntittively less ccurte thn those in Tble 6 Simpson s rule γ γ 3, nine smple points) is more ccurte thn the midpoint rule γ γ 0, one smple point) which in turn is more ccurte thn the trpezoidl rule γ γ, four smple points) We note tht the

298 3 6 DOUBLE INTEGRAL INEQUALITIES 90 γ γ Numericl Error Theoreticl Error ) 46-) ) -) ) 9-) 45-) 38-) Tble 6 The numericl nd theoreticl errors in computing 69) using 67) with x x 05 nd vrious vlues of γ, γ theoreticl errors re symmetric bout γ γ in Tble 6, but this is not the cse in Tble 6; these properties re esy to see by inspection of 6) nd 67) respectively 65 Appliction For Cubture Formule To illustrte the use of cubture formul, we form composite rule from the inequlity 65) Let us consider the rbitrry division: I n : ξ 0 < ξ < < ξ n b on the intervl, b with x i ξ i, ξ i+ for i 0,,, n nd Jm : τ 0 < τ < < τ m b on the intervl, b with y j τ j, τ j+ for j 0,,, m Consider the sum n 60) A f, I n, J m, x, y) : m j0 n h i v j f x i, y j ) n m j0 m j0 h i τ j+ v j ξi+ τ j f x i, t ) dt ξ i f t, y j ) dt, where h i ξ i+ ξ i i 0,,, n ) nd v j τ j+ τ j j 0,,, m ) nd γ γ 0 Under the bove ssumptions the following theorem holds Theorem 64 Let f :, b, b R be s in Theorem 6 nd I n, J m, x, y be s bove Then we hve the cubture formul 6) f t, t ) dt dt A f, I n, J m, x, y) + R f, I n, J m, x, y), where the reminder term R f, I n, J m, x, y) stisfies the inequlity 6) R f, I n, J m, x, y) f t,t n m j0 4 h i + x i ξ ) i + ξ i+ 4 v j + y j τ ) j + τ j+

299 9 G Hnn Proof Apply Theorem 6 on the intervl ξ i, ξ i+ τ j, τ j+, i 0,,, n ), j 0,,, m ) to get h i v j { }} ) { { }} { ξi+ ξi+ ξ i τ j+ τ j ) f x i, y j ) v j f t, y j ) dt ξ i τ j+ ξi+ τ j+ h i f x i, t ) dt + f t, t ) dt dt τ j ξ i τ j f t,t 4 h i + x i ξ ) i + ξ i+ 4 v j + y j τ ) j + τ j+ for ll i 0,,, n ), j 0,,, m ) Now, summing over i from 0 to n nd over j from 0 to m, nd using the generlized tringle inequlity, we deduce 6) Corollry 65 We know tht x i ξ i +ξ i+ h i nd Applying these to 6), we find tht R f, I n, J m, x, y) n f t,t f t,t 4 m j0 n m h i j0 y j τ j+τ j+ 4 h i + 4 h i 4 v j + 4 v j v j v j Corollry 66 Now, consider the cse where x i nd y i re the midpoints At the midpoint we hve 63) f t, t ) dt dt A f, I n, J m ) + R f, I n, J m ), where the reminder term R f, I n, J m ) stisfies R f, I n, J m ) f t,t 6 n m h i j0 Corollry 67 Let the conditions of Theorem 64 hold In ddition, let I n be the equidistnt prtition of, b, I n : x i + b ) n i, i 0,,, n, nd J m be the equidistnt prtition of, b, J m : y j + b ) m j, j 0,,, m, then 64) ft, t )dt dt Af, I n, J m ) v j f t, t b ) b ) 6nm

300 6 DOUBLE INTEGRAL INEQUALITIES 9 b n Proof From Theorem 64 with h i for ll i so tht f n t, t ) m b b Rf, I n, J m ) 6 n m j0 f t, t b ) b ) 6nm nd hence the result is proved Remrk 63 If we were to use 60) to pproximte the integrl ft, t ) dt dt with uniform grid nd smpling t ech mid-point, then the reminder R is bounded by f t, t b ) b ) 65) R f, I n, J m, x, y) 6nm n m Numericl Error Error rtio Theoreticl Error 5-3) 63-) 0-4) 45 6-) ) ) ) ) ) ) ) ) ) ) ) ) Tble 63 The numericl nd theoreticl errors in evluting 69) using the cubture rule in 60) for vrious vlues of n, m Smpling occurs t the mid-point of ech region ) Tble 63 shows the numericl nd theoreticl errors in pplying the mid-point cubture rule 60) to evlute the double integrl 69) for n incresing number of intervls The numericl error rtio suggests tht this composite rule hs convergence R O n m This contrsts with 64) which predicts convergence rte of R 6nm It should be noted tht the development of the bounds in Section 63 ssumes tht the integrnd is once differentible This condition dmits wider clss of functions thn the usul bounds for Newton-Cotes rules, but the error estimte will be more conservtive if its pplied, s it is here, to n integrnd tht is infinitely smooth In ddition, theoreticl optimlity occurs t γ γ, while numericlly this vlue seems to be γ γ 3 Becuse of the behviour of the integrnd, Simpson s rule which is optiml in the Newton-Cotes sense) for the clss of fourth differentible mppings, will be superior The methods of Section 63 cn be pplied to smoother 6 s well s weighted mppings Work is continuing in this direction )

301 93 G Hnn 66 Mpping Whose First Derivtives Belong to L p, b) For this section we will refer to where S S Drgomir nd S Wng produced some pplictions of Ostrowski s inequlity to some specil mens nd numericl qudrture rules In, the sme uthor considered nother inequlity of Ostrowski type for p norms s follows: Theorem 68 Let f : I R R be differentible ) mpping on I nd, b I with < b If f L p, b) p >, p + q, then we hve the inequlity 66) f x) b f t) dt x ) q+ + b x) q+ f b q + p for ll x, b where f p : b f t) p dt ) p is the L p, b) norm In this section we point out n inequlity of Ostrowski type for double integrls for the first differentible mpping in terms of the p norm Theorem 69 Let f : R R be differentible mpping on, b, b nd let f t,t f t t be bounded on, b ), b ), tht is, b f t p,t : f t t p dt dt ) p, nd with the sme conditions s in Theorem 6) We cn obtin the following inequlity 67) Gx, t, x, t ) f p t,t γ q+ + γ ) q+ q x ) q+ + b x ) q+ q q + ) q γ q+ + γ ) q+ q x ) q+ + b x ) q+ q,

302 6 DOUBLE INTEGRAL INEQUALITIES 94 Proof As in Theorem 6) nd by pplying Hölder s inequlity for double integrls, tht is, f 68) p x, t ) p x, t ) dt dt t t ) b q ) p x, t ) p x, t ) q dt dt p p f t t dt dt ) b q ) b q p x, t ) q dt p x, t ) q dt f p t,t Consider G x ) ) q p x, t ) q dt α ) x ) α t ) q dt + t α ) q dt α ) β b + β t ) q dt + t β ) q dt x β ) q α ) q+ + x α ) q+ + β x ) q+ + b β ) q+ q + nd we get on using 6) γ q+ + γ ) q+ x ) q+ + b x ) q+ G x ) q + Similrly, γ q+ + γ ) q+ x ) q+ + b x ) q+ G x ) q + Then substituting into 66) will produce the result 67) nd thus the theorem is proved Corollry 60 With the conditions s in Theorem 6, then f t p,t 69) Gx, t, x, t ) q γ q+ 4 q + ) + γ ) q+ q b ) q+ q q γ q+ + γ ) q+ q b ) q+ q q q Proof Plce x i i+bi in eqution 67)

303 95 G Hnn Remrk 64 If p q, then 69) becomes f t Gx, t, x, t ),t γ 3 + γ ) 3 b 3 630) γ 3 + γ ) 3 q b 3 Remrk 65 If γ γ 0, then 630) becomes b + b ) b ) f, ) + b b 63) b ) f ) + b b b ) f, t dt f t, t ) dt dt f t,t b ) b ) 3 t, + b Remrk 66 If γ γ,630) becomes b ) b ) 63) f b, b ) + f, b ) + f b, ) + f, ) 4 b ) f t, b ) dt + b ) + b ) f b, t ) dt + b ) + f t, t ) dt dt f t,t f t, ) dt f, t ) dt b ) b ) 3 ) dt Remrk 67 If γ γ, then 630) becomes 633) b ) b ) + b f, ) + b 4 + b ) b ) f b, ) + b + f, ) ) + b + b + f, 8 ) + b +f, + b ) b ) 6 f b, b ) + f, b ) + f b, ) + f, ) b ) b ) + b b b f, t dt + f b, t ) dt + f, t ) dt 4 b ) b f t, ) + b b dt + f t, b ) dt 4 b + f t, ) dt + f t, t ) dt dt f t,t b ) b ) 3 48

304 6 DOUBLE INTEGRAL INEQUALITIES 96 Remrk 68 Let f t, t ) g t ) g t ) where g :, b R If g is continuous nd stisfies the condition tht ) g b p t) p g t) p dt, b then, for x x x, we get b ) g x) g x) g x) b ) g x) b ) g p q + ) q Therefore, g t) dt b ) g x) This gives g t) dt + g t) dt g t) g t) dt dt ) x ) q+ + b x) q+ q g p q + ) q ) x ) q+ + b x) q+ ) q g t) dt b ) g x) x ) q+ + b x) q+ g p q + which is Ostrowski type inequlity for the p norm Thus, 67) is generliztion for two dimensionl integrls of the Ostrowski type for p norms As SS Drgomir nd S Wng proved in their pper Corollry 6 With the conditions s in Theorem 6, then f p t,t 634) Gx, t, x, t ) b q + ) ) b ) + q, q where Gx, t, x, t ) is s given in 6) q Proof x i i ) q+ + b i x i ) q+ b i i ) q+ nd γ q+ i + γ i ) q+ 67 Appliction For Cubture Formule Let us consider the rbitrry division: I n : ξ 0 < ξ < < ξ n b on the intervl, b with x i ξ i, ξ i+ for i 0,,, n nd Jm : τ 0 < τ < < τ m b on the intervl, b with y j τ j, τ j+ for j 0,,, m

305 97 G Hnn Consider the sum n 635) A f, I n, J m, x, y) : m j0 n n h i v j f x i, y j ) m j0 v j ξi+ m j0 ξ i f t, y j ) dt, h i τ j+ τ j f x i, t ) dt where h i ξ i+ ξ i i 0,,, n ) nd v j τ j+ τ j j 0,,, m ) Under the bove ssumptions the following theorem holds Theorem 6 Let f :, b, b R be s in Theorem 64 nd I n, J m, x, y be s bove Then we hve the cubture formul 636) f t, t ) dt dt A f, I n, J m, x, y) + R f, I n, J m, x, y), where the reminder term R f, I n, J m, x, y) stisfies the inequlity 637) R f, I n, J m, x, y) f t,t p q + ) q n m j0 x i ξ i ) q+ + ξ i+ x i ) q+ q 638) y i ζ i ) q+ + ) ) q+ f p q t,t ζ i+ y i q + ) q n m j0 h i v j ) + q Proof Apply inequlity 67 on the intervl ξ i, ξ i+ ζj, ζ j+, i 0,,, n ), j 0,,, m ) to get ) ) ξi+ ξi+ ξ i ζj+ ζ j f xi, y j ) v j f t, y j ) dt ξ i ζj+ ξi+ ζj+ h i f x i, t ) dt + f t, t ) dt dt ζ j ξ i ζ j x i ξ i ) q+ + ) q+ q ξ i+ x i y i ζ i ) q+ + ) q+ q ζ i+ y i q + ) q ζj+ ζ j ξi+ ξ i f t t p dt dt ) p for ll i 0,,, n ), j 0,,, m ) Summing over i from 0 to n nd over j from 0 to m, nd using the generlized

306 6 DOUBLE INTEGRAL INEQUALITIES 98 tringle inequlity nd Hölder s directed inequlity, we obtin R f, I n, J m, x, y) n m x q + ) i ξ i ) q+ + ) q+ q ξ i+ x i q j0 y i ζ i ) q+ + ) q+ q ζ i+ y i q + ) q n m j0 ζj+ ζ j ξi+ x i ξ i ) q+ + ξ i+ x i ) q+ q y i ζ i ) q+ + ) ) q+ q ζ i+ y i j0 n ζj+ ζ j ξ i ξi+ x i ξ i ) q+ + ) ) q ) q q+ q ξ i+ x i q + ) q m y i ζ i ) q+ + ) ) q q+ q ζ i+ y i n m j0 ζj+ ζ j ξi+ ξ i f t t p q ξ i ) p dt dt p ) f p p t t dt dt f t t p p ) p dt dt q + ) q n m j0 x i ξ i ) q+ + ξ i+ x i ) q+ q y i ζ i ) q+ + ζ i+ y i ) q+ q ) f t,t p, nd the first inequlity in 637) is proved The second prt follows directly from the fct tht x i ξ i ) q+ + ξ i+ x i ) q+ h q+ i nd y i ζ i ) q+ + ζ i+ y i ) q+ v q+ j 68 Mppings Whose First Derivtives Belong to L, b) In this section n inequlity of Ostrowski type for two dimensionl integrls for functions whose first derivtives belong to L cn be produced s shown in the following theorem,

307 99 G Hnn Theorem 63 Let f : R R be differentible mpping on, b, b nd let f t,t f t t be bounded on, b ), b ), tht is, b f t,t : f t t dt dt <, nd with the sme conditions s in Theorem 6) We cn obtin the following inequlity 639) Gx, t, x, t ) f t,t where M i b i i ) + γ 4 i nd Gx, t, x, t ) is s given in 6) M i, i + x i i + b i ) + γ i ) Proof The proof follows tht of Theorem 6) we hve, f 640) p x, t ) p x, t ) dt dt t t ) b b ) p x, t ) p x, t ) dt dt f t t dt dt sup p x, t )p x, t ) f t,t t,t ),b X,b sup p x, t ) sup p x, t ) Now, consider 64) Let nd t,b G x ) sup t,b t,b p x, t ) mx{α, x α, β x, b β } M x ) mx{α, x α } x + α + x x + + γ M x ) mx{β x, b β } b x + β b + x b x + + γ

308 6 DOUBLE INTEGRAL INEQUALITIES 300 then nd similrly G x ) mx{m, M } b 4 G x ) b 4 + γ + γ + x + b ) + γ ) + x + b ) + γ ) Substituting into 640) will produce the result in 639) nd thus the proof completed Corollry 64 With the conditions s in Theorem 635, then f t Gx, t, x, t ),t 64) b i i ) + γ 6 i i Proof Put x i i+bi in eqution 639) Remrk 69 If γ γ 0, then 64) becomes b + b ) b ) f, ) + b b 643) b ) f ) + b b b ) f, t dt f t, t ) dt dt f t,t b ) b ) 4 t, + b Remrk 60 If γ γ, then 64) becomes b ) b ) 644) f b, b ) + f, b ) + f b, ) + f, ) 4 b ) f t, b ) dt + b ) + b ) f b, t ) dt + b ) + f t, t ) dt dt f t,t b ) b ) 4 Remrk 6 If γ γ, then 64) becomes 645) b ) b ) + b f, ) + b 4 f t, ) dt f, t ) dt + b ) b ) 8 ) dt

309 30 G Hnn f b, ) + b + f, ) ) ) + b + b + b + f, + f, b + b ) b ) f b, b ) + f, b ) + f b, ) + f, ) 6 b ) b ) + b f, t dt 4 + f b, t ) dt + + f t, b ) dt + f t,t 6 f, t ) dt b ) 4 b ) b ) f t, ) dt + f f t, t ) dt dt t, ) + b dt Remrk 6 Let f t, t ) g t ) g t ) where g :, b R If g is continuous nd stisfies the condition tht ) g b t) g t) dt, b then, for x x x, nd γ we get b ) g x) g x) g x) b ) g t) dt g x) b ) g t) dt + g t) g t) dt dt g ) b ) + 4 x + b Therefore, g t) dt b ) g x) g ) b ) + 4 x + b This gives b ) g t) dt b ) g x) g + 4 x + b which is Ostrowski type inequlity for the norm Thus, 639) is generliztion for two dimensionl integrls of the Ostrowski type inequlity for norms 69 Integrl Identities In, P Cerone, SS Drgomir nd J Roumeliotis proved the following Ostrowski type inequlity for n time differentible mppings

310 6 DOUBLE INTEGRAL INEQUALITIES 30 Theorem 65 Let f :, b R be mpping such tht f n ) is bsolutely continuous on, b nd f n) L, b Then for ll x, b, we hve the inequlity: b n b x) k+ + ) k x ) k+ 646) f t) dt f k) x) k + )! k0 f n) n + )! where f n) : sup t,b x ) n+ + b x) n+ f n) t) < f n) b ) n+, n + )! For other similr results for n time differentible mppings, see the pper 7 by Fink nd 8 by Anstssiou In 3 nd 4 the uthors proved some inequlities of Ostrowski type for double integrls in terms of different norms In this section we combine the bove two results nd develop them in two dimensions to obtin generliztion of the Ostrowski inequlity for n-time differentible mppings using different types of norms The result presented here pproximtes two-dimensionl integrl for n time differentible mppings vi the ppliction of function evlutions of one dimensionl integrls t the boundry nd n interior point The following result holds Theorem 66 Let f :, b c, d R be continuous mpping such tht the following prtil derivtives l+k f, ), k 0,,, n, l 0,,, m exist nd x k y l re continuous on, b c, d Further, for K n :, b R, S m : c, d R given by t ) n n!, t, x K n x, t) : t b) n 647) n!, t x, b s c) m m!, s c, y S m y, s) : s d) m m!, s y, d then for ll x, y), b c, d, we hve the identity: 648) d n + ) m X k x) k0 c n f t, s) ds dt d c m k0 l0 S m y, s) k+m f x, s) x k s m K n x, t) n+l f t, y) t n y l dt + ) m+n X k x)y l y) l+k f x, y) x k y l d ds + )n c m l0 Y l y) K n x, t)s m y, s) n+m f t, s) t n s m ds dt,

311 303 G Hnn where 649) X k x) b x)k+ + ) k x ) k+ k+)!, Y l y) d y)l+ + ) l y c) l+ l+)! Proof Applying the identity see ) b n b x) k+ + ) k x ) k+ 650) g t) dt g k) x) k + )! where k0 + ) n P n x, t) g n) t) dt, P n x, t) t ) n n! if t, x, t b) n n! if t x, b, which hs been used essentilly in the proof of Theorem 65, for the prtil mpping f, s), s c, d, we cn write 65) f t, s) dt n b x) k+ + ) k x ) k+ k0 for every x, b nd s c, d + ) n k + )! K n x, t) n f t, s) t n dt k f x, s) x k Integrting 65) over s on c, d, we deduce d n b x) k+ + ) k x ) k+ 65) f t, s) ds dt c k + )! k0 d k f x, s) b ) d c x k ds + ) n n f t, s) K n x, t) c t n ds dt for ll x, b Applying the identity I ) gin for the prtil mpping k fx, ) x k 653) d c k f x, s) x k ds m l0 d + ) m S m y, s) m k f x, s) c s m x k m d y) l+ + ) l y c) l+ l0 l + )! d y) l+ + ) l y c) l+ l + )! ) ds l+k f x, y) x k y l + ) m on c, d, we obtin l y l d c k ) f x, y) x k S m y, s) k+m f x, s) x k s m ds

312 6 DOUBLE INTEGRAL INEQUALITIES 304 In ddition, the identity 650) pplied for the prtil derivtive n ft, ) t lso gives n d n m t, s) d y) l+ + ) l y c) l+ n+l f t, y) 654) t n ds l + )! t n y l c l0 + ) m d c S m y, s) n+m f t, s) t n s m ds Using 653) nd 654) nd substituting into 65) will produce the result 648), nd thus the theorem is proved Corollry 67 With the ssumptions s in Theorem 66, we hve the representtion d n m ) ) + b c + d l+k f +b f t, s) ds dt X k Y, ) c+d 655) l c x k y l k0 l0 n ) + ) m + b d X k S m s) k+m f +b, s) k0 c x k s m ds m ) + ) n c + d b Y l K n t) n+l f ) t, c+d t n y l dt l0 + ) m+n d c K n t) S m s) n+m f t, s) t n s m ds dt, where X k ) nd Y l ) re s given in 648) nd so ) + b + ) k b ) k+ X k k + )! k+, ) c + d + ) l d c) l+ Y l l + )! l+, nd K n :, b R, S m : c, d R re given by ) + b K n t) K n, t nd on using 647) ) c + d S m s) S m, s Corollry 68 Let f be s in Theorem 66 Then we hve the following identity 656) d c f t, s) ds dt 4 n m k0 l0 b ) k+ d c) l+ k + )! l + )! l+k x k y l f, c) + ) l f, d) + ) k f b, c) + ) l+k f b, d) + n 4 )m k0 b ) k+ k + )! d c Y m s) k+m x k s m f, s) + ) k f b, s) ds

313 305 G Hnn + m 4 )n + 4 l0 d c d c) l+ b X n t) n+l l + )! t n y l f t, c) + ) k f t, d) dt X n t) Y m s) n+m f t, s) t n s m ds dt, where X n t) nd Y m s) re s given by 649) Proof By substituting x, y), c),, d), b, c), b, d) respectively nd summing the resulting identities nd fter some simplifiction, we get the desired inequlity 656) We strt with the following result 60 Some Integrl Inequlities Theorem 69 Let f :, b c, d R be continuous on, b c, d, nd ssume tht n+m f t n s exist on, b) c, d) Then we hve the inequlity m d n m 657) f t, s) ds dt X k x) Y l y) l+k f x, y) x k y l n ) m X k x) k0 n+)!m+)! c d c k0 l0 Sy, s) k+m f x, s) x k s m x ) n+ + b x) n+ ds )n m l0 Y l y) Kx, t) n+l f t, y) t n y l dt y c) m+ + d y) m+ n+m f t n s m if n!m! if n+m f t n s m L, b c, d) ; x ) nq+ +b x) nq+ nq+ q y c) mq+ +d y) mq+ mq+ n+m f t n s m L p, b c, d), p >, p + q ; q n+m f p t n s m 4n!m! x )n + b x) n + x ) n b x) n y c) m + d y) m + y c) m d y) m if n+m f t n s L m, b c, d) for ll x, y), b c, d, where n+m f t n s m sup n+m f t n s m p t,s),b c,d d c n+m f t, s) t n s m <, n+m p t n f t, s) sm dtds n+m f t n s m ) p <

314 6 DOUBLE INTEGRAL INEQUALITIES 306 Proof Using Theorem 66, we get from 648) 658) d n m f t, s) ds dt X k x) Y l y) l+k f x, y) x k y l c k0 l0 n ) m k0 X k x) d c d c d c Sy, s) k+m f x, s) x k s m ds )n m l0 Y l y) K n x, t) S m y, s) n+m f t, s) t n s n ds dt K n x, t) S m y, s) n+m f t, s) t n s n ds dt Kx, t) n+l f t, y) t n y l dt Using Hölder s inequlity nd properties of the modulus nd integrl, then we hve tht 659) d c K n x, t) S m y, s) n+m f t, s) t n s m ds dt n+m f t n s m n+m f p t n s m p >, p + q ; n+m f t n s m d c K n x, t) S m y, s) dt ds ) d c K n x, t) S m y, s) q q dt ds, sup t,s),b c,d K n x, t) S m y, s) Now, from 659) nd using 647), d K n x, t) S m y, s) dt ds K n x, t) dt S m y, s) ds c c x t ) n b b t) n y s c) m d d s) m dt + dt ds + ds n! x n! c m! y m! x ) n+ + b x) n+ y c) m+ + d y) m+ n + )! m + )! d giving the first inequlity in 657)

315 307 G Hnn Further, on using 647) nd from 659) d c K n x, t) S m y, s) q ds dt x t ) nq dt + n!m! n!m! nq + x ) nq+ + b x) nq+ x ) q b t) nq dt q K n x, t) q dt q ) q d y s c) mq ds + c c d y c) mq+ + d y) mq+ mq + y S m y, s) q ds dt d s) mq ds q q ) q producing the second inequlity in 657) Finlly, from 647) nd 659), sup t,s),b c,d { n x ) mx, n! K n x, t) S m y, s) sup } b x)n mx n! t,b { m y c), m! K n x, t) sup s c,d d y)m m! } S m y, s) x ) n + b x) n + x ) n b x) n n!m! y c) m + d y) m + y c) m + d y) m gives the inequlity in 657) where we hve used the fct tht mx {X, Y } X + Y + Y X Thus the theorem is now completely proved From the results of Theorem 69 bove, we hve the following corollry

316 6 DOUBLE INTEGRAL INEQUALITIES 308 Corollry 60 With the ssumptions of Theorem 69, we hve the inequlity d 660) f t, s) ds dt c n m k0 l0 n ) m ) ) + b c + d l+k + b X k Y l x k y l f, c + d ) k0 m ) n l0 ) + b d X k S m s) c ) c + d b Y l k+m ) + b x k s m f, s ds K n t) n+l t n y l f t, c + d ) dt n+m f t n s ; m n+m n+)!m+)! b )n+ d c) m+ n+m n!m! b ) nq+ d c) mq+ nq+)mq+) q n+m f p t n s ; m n+m n!m! b )n d c) m n+m f t n s, m where p p, ) re the Lebesgue norms on, b c, d Proof Tking x +b nd y c+d in 657) redily produces the result s stted These re the tightest possible for their respective Lebesgue norms, becuse of the symmetric nd convex nture of the bounds in 657) Remrk 63 For n m in 660) nd f Lebesgue spces on, b c, d, we hve d 66) f t, s) ds dt b ) d c) f c t s + b, c + d ) d + b ) S s) ) + b b c s f, s ds + d c) 66) 6 b ) d c) b ) q+ d c) q+ q 4 q+) f t s belonging to the pproprite ; f t s 4 b ) d c) f t s, K t) t f t, c + d ) dt ; p nd thus some of the results of 5 nd 6 re recptured

317 309 G Hnn Corollry 6 With the ssumptions on f s outlined in Theorem 69, we cn obtin nother result which is generliztion of the Trpezoid inequlity 663) d c n f t, s) ds dt m k0 l0 b ) k+ k + )! d c)l+ l + )! f, c) + ) l f, d) + ) k f b, c) + ) k+l f b, d) 4 n ) m k0 m ) n l0 b ) k+ k + )! κ n,m b ) n+ d c) m+ n+)!m+)! d c Y l s) d c) l+ X k t) l+n l + )! t n y l n+m f t n s ; m k+m f, s) + ) k f b, s) x k s m ds 4 f t, c) + ) l f t, d) dt 4 if n+m f t n s m L, b c, d) ; n+m f p ) b t n s m T n, b; t) q q d dt if c T m c, d; s) q ds n+m f t n s m L p, b c, d), p >, p + q ; b ) n d c) m 4n!m! n+m f t n s, m ) q if n+m f t n s L m, b c, d) where κ n,m : if n r nd m r, n n if n r + nd m r, m m if n r nd m r +, n ) n m ) m if n r + nd m r +

318 6 DOUBLE INTEGRAL INEQUALITIES 30 Proof Using the identity 656), we find tht d c l+k x k y l n f t, s) ds dt n ) m k0 m k0 l0 b ) k+ k + )! d c)l+ l + )! f, c) + ) l f, d) + ) k f b, c) + ) k+l f b, d) 4 b ) k+ k + )! d c s c) m + s d) m m! k+m f, s) + ) k f b, s) x k s m ds 4 m ) n d c) l+ b t ) n + t b) n n+l f t, c) + ) l f t, d) l + )! l0 n! y l t n dt 4 d T n, b; t) T m c, d; s) n+m f c t n s m dsdt n+m f d t n s m c T n, b; t) T m c, d; s) dt ds if n+m f t n s m L, b c, d) ; n+m f p ) b t n s m T n, b; t) q q d dt c T m c, d; s) q ds if n+m f t n s L m p, b c, d), p >, p + q ; n+m f t n s sup T m n, b; t) T m c, d; s) t,s),b c,d ) q if n+m f t n s m L, b c, d) where T n, b; t) b t) n + ) n t ) n, n! T m c, d; s) d s) m + ) m s c) m m! Now consider T n, b; t) dt As my be seen, explicit evlution of the integrl depends on whether n is even or odd

319 3 G Hnn i) If n is even, put n r Therefore, T n, b; t) dt r )! r )! b t) r + t ) r dt b ) r+ + r + b )r+ r + d c Similrly, T m c, d; s) ds r )! b )r+ r + )! d c b )n+ n + )! d s) r + s c) r ds ii) Now, if n is odd, tht is, n r +, then d c)m+ m + )! T n, b; t) b t)r+ t ) r+ r + )! Let g t) b t) r+ t ) r+ We cn observe tht g t) < 0 for ll t +b, b g t) 0 t t +b g t) > 0 for ll t, +b ) Thus r + )! T n, b; t) dt +b b t) r+ t ) r+ dt + b )r+ r + 4 b ) r+ r + +b t ) r+ b t) r+ dt nd so Similrly, T n, b; t) dt d c b ) r+ r + ) r + )! r+ b )r+ r + )! T m c, d; s) ds nd this gives the first inequlity in 663) r + r+ d c)m+ m + )! m b )n+ n + )! m n n

320 6 DOUBLE INTEGRAL INEQUALITIES 3 Now, for the third inequlity we hve, sup T n, b; t) t,b n! sup b t) n + t ) n ) b )n n! for ll n even t,b sup b t) n t ) n b )n n! for ll n odd t,b nd this gives lst prt of the inequlity in 663) The corollry is thus completely proved Remrk 64 For n m, we hve tht d c b f t, s) ds dt + b ) d c) 4 d f, s) + f b, s)) ds c f, c) + f, d) + f b, c) + f b, d) d c f t, c) + f t, d)) dt b ) d c) 4 b )d c)) q+ q 4 q+) b )d c) 4 x ) + b x) y c) + d y) f t s f t s, p >, p + q ; p f t s Agin, the sme result ws obtined by G Hnn et l in 5 nd S Drgomir et l in 6 6 Applictions to Numericl Integrtion The following ppliction in Numericl Integrtion is nturl to be considered Theorem 6 Let f :, b c, d R be s in Theorem 69 In ddition, let I v nd J µ be rbitrry divisions of, b nd c, d respectively, tht is, I v : ξ 0 < ξ < < ξ ν b, where x i ξ i, ξ i+ ) for i 0,,, ν, nd J µ : c τ 0 < τ < < τ µ d,

321 33 G Hnn with y j τ j, τ j+ ) for j 0,,, µ, then we hve the cubture formul 664) d c n f t, s) ds dt n ν µ + ) m m + ) n k0 j0 l0 ν µ j0 +R f, I v, J µ, x, y), m ν µ k0 l0 j0 τ j+ S j) X i) k x i) τ j ξi+ Y j) l y j ) K i) ξ i where the reminder term stisfies the condition Rf, I n, J m, x, y) X i) k x i)y j) l y j ) i+j f x i, y j ) x i y j m y j, s) k+m f x i, s) x k s m ds n x i, t) n+l f t, y j ) t n y l dt n+m f t n s m n+)!m+)! ν µ j0 x i ξ i ) n+ + ξ i+ x i ) n+ y j τ j ) m+ + τ j+ y j ) m+ if n+m f t n s m L, b c, d) ; where X i) k nd n+m f p t n s m n!m!nq+) q n+m f t n s m ν 4n!m! µ j0 ν x i ξ i ) nq+ + ξ i+ x i ) nq+ q µ j0 y j τ j ) mq+ + τ j+ y j ) mq+ q if n+m f t n s m L p, b c, d), p >, p + q ; xi ξ i ) n + ξ i+ x i ) n + x i ξ i ) n ξ i+ x i ) n y j τ j ) m + τ j+ y j ) m + y j τ j ) m τ j+ y j ) m if n+m f t n s m L, b c, d) ; j) k 0,, n ; i 0,, ν ), Y l l 0,, m ; j 0,, µ ) K i) n i 0,, ν ), S m j) j 0,, µ ) re defined by ) k+ X i) k x ξi+ x i + ) k x i ξ i) : i ) k+, k + )! Y j) l y j ) : τ j+ y j ) l+ + ) l y j τ j ) l+, l + )! t ξ i ) n K n i) n!, t ξ i, x i x i, t) : t ξ i+ ) n n!, t x i, ξ i+

322 6 DOUBLE INTEGRAL INEQUALITIES 34 nd S m j) y j, s) : s τ j) m m!, s τ i, y i s τ j+) m m!, s y i, τ j+ The proof is obvious by Theorem 69 pplied on the intervl ξ i, ξ i+ τ j, τ j+, i 0,, ν ; j 0,, µ ), nd we omit the detils Remrk 65 Similr result cn be obtined if we use the other results obtined in Section 63, but we omit the detils

323 Bibliogrphy S S DRAGOMIR nd S WANG; Applictions of Ostrowski s inequlity to the estimtion of error bounds for some specil mens nd some numericl qudrture rules Appl Mth Lett, 998), S DRAGOMIR nd S WANG; A new inequlity of Ostrowski s type in L p norm, Accepted 3 DS MITRINOVIĆ, JE PEČARIĆ nd AM FINK, Inequlities for functions nd their Integrls nd Derivtives, Kluwer Acdemic, Dordrecht, N S BARNETT nd S S DRAGOMIR, An Ostrowski type inequlity for Double Integrls nd Appliction for Cubture formule Submitted to Soochow J of Mth, 999 ONLINE Avilble online from the RGMIA: rgmi/vnhtml 5 P CERONE nd SS DRAGOMIR, Three Point Qudrture Rules Involving, t most, First Derivtive Submitted to SIAM J Num Anl, 999 ONLINE Avilble online from the RGMIA: rgmi/vn4html 6 P CERONE, SS DRAGOMIR nd J ROUMELIOTIS, Some Ostrowski Type Inequlities for n-time Differentible Mppings nd Applictions Accepted for publiction in Demonstrtio Mthemtic, 999 ONLINE Avilble online from the RGMIA: rgmi/vnhtml 7 M GOLOMB nd HF WEINBERGER, Optiml Approximtion nd Error Bounds, in On Numericl Approximtion RE Lnger, ed), pp7-90, University of Wisconsin Press, Mdison, D S MITRINOVIĆ, J E PE CARIĆ nd A M FINK, Inequlities Involving Functions nd Their Integrls nd Derivtives, Dordrecht: Kluwer Acdemic, 99 9 J ROUMELIOTIS, P CERONE nd SS DRAGOMIR, An Ostrowski Type Inequlity for Weighted Mppings with Bounded Second Derivtives Accepted for publiction in J KSIAM, 999 ONLINE Avilble online from the RGMIA: rgmi/vnhtml 0 JF TRAUB nd H WOŹNIAKOWSKI, A Generl Theory of Optiml Algorithms, Acdemic Press, 980 P CERONE, SS DRAGOMIR nd J ROUMELIOTIS, Some Ostrowski type inequlities for n time differentible mppings nd pplictions, Demonstrtio Mth, 3 999), No 4, DS MITRINOVIĆ, JE PEČARIĆ nd AM FINK, Inequlities for Functions nd their Integrls nd Derivtives, Kluwer Acdemic, Dordrecht, N S BARNETT nd S S DRAGOMIR, An Ostrowski type inequlity for double integrls nd ppliction for cubture formule, Soochow J of Mth, in press) 4 S S DRAGOMIR, N S BARNETT nd P CERONE, An Ostrowski type inequlity for double integrls in terms of L p- Norms nd pplictions in Numericl integrtions, Anl Num Theor Approx Romni), ccepted) 5 G HANNA, P CERONE nd J ROUMELIOTIS, An Ostrowski type inequlity in two dimensions using the three point rule, CTAC Computtionl Techniques nd Applictions Conference 6 S S DRAGOMIR, P CERONE, N S BARNETT nd J ROUMELIOTIS, An inequlity of the Ostrowski type for double integrls nd pplictions for cubture formule, submitted 7 AM FINK, Bounds on the derivtion of function from its verges, Czechoslovk Mth J, 4 99),

324 BIBLIOGRAPHY 36 8 GA ANASTASSIOU, Ostrowski type inequlities, Proc of the Americn Mth Soc, 3 995), No,

325 CHAPTER 7 Product Inequlities nd Weighted Qudrture by J ROUMELIOTIS Abstrct Weighted or product) integrl inequlities re developed vi Ostrowski nd Grüss pproches The inequlities provide n error estimte for weighted integrls where both the qudrture rule nd error bound re given in terms of t most) the first three moments of the weight Rule type is distinguished nd interior point, boundry point nd three point rules re explored Results for the most populr weight functions re tbulted Numericl experiments re provided nd comprisons with other product rules of similr order re mde The methods outlined in this chpter llow for the genertion of non-uniform qudrture grids with respect to ny rbitrry weight employing only smll number of weight moments 7 Introduction The Ostrowski inequlity 3 is very fruitful strting point for the development of numericl integrtion rules From this point of view, one of the more importnt spects of the inequlity nd of Montgomery s identity) is its role in the production of bounds for the well known Newton-Cotes rules To illustrte this point, consider the integrl 7) I fx) dx, where f is some bounded function defined on the finite intervl, b If b is smll, we my pproximte I by smpling t one point 7) I x) b )fx), for some x b Using Ostrowski-type results, the error I I x) hs been tbulted for vriety of properties of f For exmple, if f exists nd is bounded then we cn show 3 tht 73) I I x) b ) x +b b ) ) ) f

326 7 PRODUCT INEQUALITIES AND WEIGHTED QUADRATURE 38 Eqution 73) is the well known Ostrowski inequlity If f is integrble 0, 3 or if f hs bounded vrition 5 or is L-Lipschitzin 7, 36 then we hve respectively, b ) 74) I I x) b ) + + b ) 4 + x +b b x +b b x +b b ) ) f, ) b f), ) Other generliztions to both the rule 7) nd the bound exploiting other properties of f for exmple smoothness, monotonicity nd convexity) hve been reported by Cerone, Drgomir, Fink, Milovnović, Pe crić nd others; see, 9, 0,, 8, 3, 6,, 8, 8, 9, 30, 34, 35 nd the book 3 From qudrture view point, the techniques described bove provide explicit priori error bounds in vriety of norms, but they fil to ccount for singulr integrnds nd/or infinite regions Integrls of this type re of prticulr importnce since they rise nturlly in the context of sttisticl estimtions for eg 4, 3), integrl equtions for eg 6, 38, Chpter 3), especilly s they impct mthemticl models for eg 5, 40, 38) In this chpter we develop weighted or product) Ostrowski nd Grüss type inequlities where the upper bound is function of the first few derivtives of the mpping We investigte smpling t interior points mid-point type), boundry points trpezoidl type) nd combintion of both three point type) The rules thus furnished provide explicit priori bounds for n rbitrry mesh rrngement The bounds cn be used to produce n optiml mesh with the respect to n rbitrry weight s well s the construction of Kronrod type rules This pproch contrsts to tht commonly used for mesh refinement, where successive posteriori comprisons re mde to obtin desired ccurcy ) L 7 Weight Functions In the following sections weighted integrl inequlities re developed We ssume tht the weight w is integrble nd non-negtive The domin my be finite or infinite nd w my vnish t the boundry points The results will be expressed in terms of the first few moments of w Definition Let w :, b) 0, ) be integrble, ie denote the first three moments to be m, M nd N, where 75) m, b) respectively wt) dt, M, b) twt) dt nd N, b) We lso introduce the following generic mesures of w wt) dt < We t wt) dt,

327 39 J Roumeliotis Definition 3 Given the conditions in Definition, the men nd vrince on the sub-intervl α, β), b) re defined s 76) nd 77) respectively σ α, β) µα, β) Mα, β) mα, β) Nα, β) mα, β) µ α, β), 73 Weighted Interior Point Integrl Inequlities Mitrinović et l 3 hve reported weighted multi-dimensionl nlogue of the Ostrowski inequlity in the first prtil derivtives of the mpping The nlysis in this chpter will be restricted to one dimension nd we begin with the result in 3 Theorem 7 3) Let w be s defined in Definition nd let f :, b R be bsolutely continuous nd hve bounded first derivtive, then 78) wt)ft) dt m, b)fx) f x t wt) dt f { x m, x) mx, b) ) + Mx, b) M, x) } Proof Define the mpping K, ) :, b R by { m, t), t, x, 79) Kx, t) mb, t), t x, b, where m is the zero-th moment s defined in 75) Integrtion by prts gives Kx, t)f t) dt x m, t)f t) dt + Producing the product Montgomery identity 70) x mb, t)f t) dt m, t)ft) x t + mb, t)ft)b tx wt)ft) dt Kx, t)f t) dt m, b)fx) Tking the modulus nd using Hölder s inequlity gives Kx, t)f t) dt f Kx, t) dt 7) { x f m, t) dt + wt)ft) dt x mt, b) dt }

328 7 PRODUCT INEQUALITIES AND WEIGHTED QUADRATURE 30 The lst result being obtined by using the fct tht for fixed x, K is positive in t, x) nd negtive in t x, b) Mking use of 70), reversing the order of integrtion in 7) nd evluting the inner integrls produces the desired result 78) Remrk 7 Substituting w into 78) returns the Ostrowski inequlity 73) Remrk 7 Unlike the Ostrowski-type results outlined in Section 7, Theorem 7 is vlid depending on w) for infinite regions nd singulr w For exmple, substituting specific weights into 78) produces the following weighted inequlities ft) ln/t) dt fx) f x ln/x) + 3/x x + /4 ), x 0,, ft) ) dt fx) t f 8/3x 3/ x + /3, x 0,, ft)e t dt fx) f e x + x ), x 0, ft)e t dt πx πfx) f erfx) + e x ), x R Corollry 7 The bound in 78) is minimized t the medin, x x where 7) m, x ) mx, b) Thus, the following medin point inequlity holds 73) wt)ft) dt m, b)fx ) f { Mx, b) M, x ) } Proof Let F represent the bound in 78) Tht is F x) x Differentiting F twice gives F x) x wt) dt F x) wx) 0, x t wt) dt x t)wt) dt + x x t x)wt) dt wt) dt m, x) mx, b) x, b) Inspection of the second derivtive revels tht the bound is convex nd hence the minimum will occur t the sttionry point From F bove, we cn see tht the minimum will occur t the medin s in eqution 7) Eqution 7) cn be used to find the optiml smpling point for the weighted inequlities in Remrk 7 The bounds in Theorem 7 nd Corollry 7 both require the second moment, M In the next two results, we present bounds tht do not rely on M but, s result, re corser thn 78) nd only pply for finite intervls nd

329 3 J Roumeliotis Corollry 73 Let the conditions in Theorem 7 hold nd let x, b, where, b is finite intervl The following product integrl inequlity holds b 74) wt)ft) dt m, b)fx) f m, b) + x + b ) Proof Observe tht x t wt) dt sup x t m, b) t,b) mx{x, b x}m, b) m, b) x ) + b x) x ) b x) ) b m, b) + x + b ) Substituting the inequlity bove into eqution 78) furnishes the desired result 74) Another estimtion in terms of the p norm of w is given in the following corollry Corollry 74 Under the bove ssumptions for f nd w nd w L p, b, we hve the inequlity 75) x ) wt)ft) dt m, b)fx) f q+ + b x) q+ /q w p, q + for ll x, b, p > nd /p+/q The bound is minimized t the mid-point x + b)/ Proof Using Hölder s inequlity we hve x t wt) dt w p x t q dt ) /q x w p x t) q dt + w p x ) q+ + b x) q+ q + x t x) q dt /q To show tht the bound is minimized t the mid-point, observe tht x ) q+ vnishes t x nd monotoniclly increses while b x) q+ vnishes t x b nd monotoniclly decreses Thus the sum is convex nd hs minimum t x ) q+ b x) q+ or x + b)/ /q The estimtion in 78) my be bounded using other properties of the mpping f For exmple if f is integrble we hve the following theorem

330 7 PRODUCT INEQUALITIES AND WEIGHTED QUADRATURE 3 Theorem 75 Let w be s given in Definition 75 nd let f :, b R be such tht f L, b) The following inequlity holds 76) wt)ft) dt m, b)fx) { f m, b) + m, x) mx, b) } The bound is minimized t the medin Proof From 70), we cn see tht wt)ft) dt m, b)fx) Kx, t)f t), dt sup Kx, t) t,b) f t) dt mx{m, x), mx, b)} f f { m, b) + m, x) mx, b) } The lst line being obtined from the well known result: mx{a, B} /A + B + A B ) Remrk 73 Substituting w into 76) returns 74), the L vrint of the Ostrowski inequlity 0, 3 Recently, Pechey et l 33 were ble to show tht the constnt / is the best possible in eqution 74) It is still n open question whether m, b) is the best possible constnt in 76) Drgomir et l 9 generlized 78) nd developed weighted Ostrowski-type inequlity for Hölder type mppings Theorem 76 9) Let w be s given in Definition nd let f be of r H Hölder type Tht is 77) fx) fy) H x y r, for ll x, y, b), H > 0 nd r 0, If wf is integrble, then the following estimtion of the product integrl holds 78) wt)ft) dt m, b)fx) H x t r wt) dt, for ll < x < b 79) Proof From eqution 77) it is esy to see tht wt) ft) fx) dt H x t r wt) dt In ddition, using the usul properties of definite integrls we cn show tht 70) wt) ft) fx) ) dt wt) ft) fx) dt Thus, combining 79) nd 70) we obtin 78) s required

331 33 J Roumeliotis Remrk 74 If in 77) r, then f is Lipschitzin If the Lipschitz constnt is L, sy, then we hve 7) wt)ft) dt m, b)fx) L x t wt) dt L { x m, x) mx, b) ) + Mx, b) M, x) } Up to this point, we hve only ssumed very generl properties regrding the mpping f nd its derivtive f For the reminder of this section, we will expnd on this somewht nd present nlogous product integrl inequlities where f is ssumed to exist An ppliction in numericl integrtion in lso included Generliztions to higher derivtives hve been mde for non-product inequlities in 3,, 34, It should be possible to generlize product integrl inequlities to higher derivtives of f if these techniques re combined with the mteril presented below Theorem 77 39) Let w be s given in Definition nd let f :, b) R hve bounded second derivtive Then the following inequlities hold 7) wt)ft) dt m, b)fx) + m, b) x µ, b) ) f x) f m, b) x ) µ, b) + σ, b), f L, b f m, b) m, x) ) mx, b) ) x µ, x) + µx, b) x + x µ, b), f L, b m, b) m, b) for ll x, b Proof Define the mpping K, ) :, b R by { t t u)wu) du, t x, 73) Kx, t) : t t u)wu) du, x < t b b Integrting by prts gives Kx, t)f t) dt x t f x) providing the identity 74) t u)wu)f t) dudt + x t Kx, t)f t) dt x u)wu) du t u)wu)f t) dudt wt)ft) dt + f x) t x b t x b t u)wu)f t) dudt t u)wu)f t) dudt x u)wu) du fx) wu) du wt)ft) dt m, b)fx)+m, b) x µ, b) ) f x)

332 7 PRODUCT INEQUALITIES AND WEIGHTED QUADRATURE 34 tht is vlid for ll x, b Now tking the modulus of 74) we hve, 75) wt)ft) dt m, b)fx) + m, b) x µ, b) ) f x) Kx, t)f t) dt f Kx, t) dt f x f t t u)wu) dudt + x t) wt) dt t x b t u)wu) dudt The lst line being computed by reversing the order of integrtion nd evluting the inner integrls To obtin the desired result 7) observe tht 76) x x t) ) wt) dt m, b) µ, b) + σ, b) Closer inspection of the kernel 73) is required in the proof of 7) Exmintion of the derivtive d Kx, t) dt { t wu) du, t, x, t wu) du, b t x, b, revels tht K increses in the intervl, x nd decreses in the intervl x, b Since K is positive, we cn deduce tht the mximum is ttined t t x Thus from 74) Kx, t)f t) dt sup Kx, t) f t,b) mx{ x x t)wt) dt, x t x)wt) dt} f Simplifying the bove expression provides the required result 7) Note lso tht 7) is vlid even for unbounded w or intervl, b The inequlity 7) is bounded in terms of three moments of w The following provides corser upper bound using only the zero-th moment

333 35 J Roumeliotis Corollry 78 Let the conditions in Theorem 77 hold The following product integrl inequlity holds 77) wt)ft) dt m, b)fx) + m, b) x µ, b) ) f x) f m, b) x + b + b ) Proof To obtin 77) note tht 78) x t) wt) dt sup x t) m, b) t,b mx{x ), x b) }m, b) x ) + x b) + x ) x b) ) m, b) x + b + b ) m, b) which upon substitution into 75) furnishes the result Corollry 79 The following inequlity for density defined on finite intervl holds Let w be density with not necessrily unit re) nd let µ nd σ be the men nd vrince respectively Then 79) for ll x, b x µ, b) ) + σ, b) x + b + b Proof The result is immeditely obvious when the identity 76) is substituted into 78) Remrk 75 Substituting x µ, b) nd x +b into 79) nd dding produces the following inequlity for the vrince in terms of the men of density 730) σ, b) b ) b + + b µ, b) A tighter bound is obtined by using 79) only once Substituting x +b gives ) σ b ) + b, b) µ 4 b ) 73) µ )b µ) 4 Corollry 70 The inequlity 7) is minimized t x µ, b) producing the men-point inequlity 73) wt)ft) dt m, b)fµ, b)) f σ, b)m, b) )

334 7 PRODUCT INEQUALITIES AND WEIGHTED QUADRATURE 36 Proof Substituting µ, b) for x in 7) produces the desired result Note tht x µ, b) not only minimizes the bound of the inequlity 7), but lso cuses the derivtive term to vnish The optiml point µ, b) cn be interpreted in mny wys For exmple, this point cn be viewed s tht which minimizes the error vrince for the probbility density w see 4 for n ppliction) This point is lso the Guss node point for one-point rule Two Interior Points Here two point nlogy of 7) is developed where the result is extended to crete n inequlity with two independent prmeters x nd x This is minly used in subsection 733 to find n optiml grid for composite weighted-qudrture rules Theorem 7 Let the conditions of Theorem 77 hold, then the following two interior point inequlity is obtined 733) wt)ft) dt m 0, ξ)fx ) + m 0, ξ) x µ, ξ) ) f x ) m 0 ξ, b)fx ) + m 0 ξ, b) x µξ, b) ) f x ) f { x m 0, ξ) µ, ξ) ) + σ, ξ) x + m 0 ξ, b) µξ, b) ) } + σ ξ, b) for ll x < ξ < x b Proof Define the mpping K,,, ) :, b 4 R by t t u)wu) du, t x, t Kx, x, ξ, t) : ξ t u)wu) du, x < t, ξ < x, t b t u)wu) du, x t b With this kernel, the proof is lmost identicl to tht of Theorem 77 Integrting by prts produces the integrl identity 734) Kx, x, ξ, t)f t) dt wt)ft) dt m 0, ξ)fx ) + m 0, b) x µ, ξ) ) f x ) m 0 ξ, b)fx ) + m 0 ξ, b) x µξ, b) ) f x ) Re-rrnging nd tking bounds produces the result 733) Corollry 7 The optiml loction of the points x, x nd ξ stisfy 735) x µ, ξ), x µξ, b), ξ µ, ξ) + µξ, b)

335 37 J Roumeliotis Proof By inspection of the right hnd side of 733) it is obvious tht choosing 736) x µ, ξ) nd x µξ, b) minimizes this quntity To find the optiml vlue for ξ write the expression in brces in 733) s x Kx, x, ξ, t) dt m 0, ξ) µ, ξ) ) + σ, ξ) x + m 0 ξ, b) µξ, b) ) 737) + σ ξ, b) ξ x t) wt) dt + ξ x t) wt) dt Substituting 736) into the right hnd side of 737) nd differentiting with respect to ξ gives d b Kµ, ξ), µξ, b), ξ, t) dt µξ, b) µξ, ) ) ) µ, ξ) + µξ, b) ξ wξ) dξ Assuming wξ) 0, then this eqution possesses only one root A minimum exists t this root since 737) is convex, nd so the corollry is proved Eqution 735) shows not only where smpling should occur within ech subintervl ie x nd x ), but how the domin should be divided to mke up these subintervls ξ) 73 Some Weighted Integrl Inequlities Integrtion with weight functions re used in countless mthemticl problems Two min res re: i) pproximtion theory nd spectrl nlysis nd ii) sttisticl nlysis nd the theory of distributions In this subsection eqution 7) is evluted for the more populr weight functions The optiml point is esily identified 73 Uniform Legendre) Substituting wt) into 76) nd 77) gives 738) µ, b) t dt b dt + b nd σ, b) t dt dt ) + b f b b ) respectively Substituting into 7) produces the second derivtive vrint of the Ostrowski inequlity 9 ft) dt b )fx) + b ) x + b ) f x) b ) + x + b ) )

336 7 PRODUCT INEQUALITIES AND WEIGHTED QUADRATURE Logrithm This weight is present in mny physicl problems; the min body of which exhibit some xil symmetry Specil logrithmic rules re used extensively in the Boundry Element Method populrized by Brebbi see for exmple 6) With wt) ln/t), 0, b, 76) nd 77) re µ0, ) 0 t ln/t) dt 0 ln/t) dt 4 nd σ 0, ) 0 t ln/t) dt 0 ln/t) dt respectively Substituting into 7) gives 0 ) ln/t)ft) dt fx) + x ) f x) 4 f The optiml point x ) ) 4 x µ0, ) 4 is closer to the origin thn the midpoint 738) reflecting the strength of the log singulrity 733 Jcobi Substituting wt) / t, 0, b into 76) nd 77) gives 0 t dt µ0, ) 0 / t dt 3 nd σ 0, ) 0 t ) t dt 0 / t dt respectively Hence, the inequlity for Jcobi weight is The optiml point 0 ft) dt fx) + x ) f x) t 3 f x ) ) 3 x µ0, ) 3 is gin shifted to the left of the mid-point due to the t / singulrity t the origin

337 39 J Roumeliotis 734 Chebyshev The men nd vrince for the Chebyshev weight wt) / t,, b re µ, ) t/ t dt / t dt 0 nd σ, ) t t dt / t dt 0 respectively Hence, the inequlity corresponding to the Chebyshev weight is The optiml point ft) dt πfx) + πxf x) t f π ) + x x µ, ) 0 is t the mid-point of the intervl reflecting the symmetry of the Chebyshev weight over its intervl 735 Lguerre The conditions in Theorem 77 re not violted if the integrl domin is infinite The Lguerre weight wt) e t is defined for positive vlues, t 0, ) The men nd vrince of the Lguerre weight re µ0, ) 0 te t dt 0 e t dt nd respectively σ 0, ) 0 0 t e t dt e t dt The pproprite inequlity is e t ft) dt fx) + x )f x) f + x ) ), 0 from which the optiml smple point of x my be deduced 736 Hermite Finlly, the Hermite weight is wt) e t defined over the entire rel line The men nd vrince for this weight re dt µ, ) te t dt 0 e t nd σ, ) t e t dt 0 dt e t respectively

338 7 PRODUCT INEQUALITIES AND WEIGHTED QUADRATURE 330 The inequlity from Theorem 77 with the Hermite weight function is thus e t ft) dt πfx) + ) πxf x) π f + x, which results in n optiml smpling point of x Appliction in Numericl Integrtion Define grid I n : ξ 0 < ξ < < ξ n < ξ n b on the intervl, b, with x i ξ i, ξ i+ for i 0,,, n The following qudrture formule for weighted integrls re obtined Theorem 73 Let the conditions in Theorem 77 hold The following weighted qudrture rule holds 739) where nd wt)ft) dt Af, ξ, x) + Rf, ξ, x) n Af, ξ, x) hi fx i ) h i x i µ i )f x i ) ) 740) Rf, ξ, x) f The prmeters h i, µ i nd σ i respectively n re given by xi µ i ) + σ i hi h i m 0 ξ i, ξ i+ ), µ i µξ i, ξ i+ ), nd σ i σ ξ i, ξ i+ ) Proof Apply Theorem 77 over the intervl ξ i, ξ i+ with x x i to obtin ξi+ wt)ft) dt h i fx i ) + h i x i µ i )f x i ) ξ i f h i xi µ i ) + σ ) i Summing over i from 0 to n nd using the tringle inequlity produces the desired result Corollry 74 The optiml loction of the points x i, i 0,,,, n, nd grid distribution I n stisfy 74) x i µ i, i 0,,, n nd ξ i µ i + µ 74) i, i,,, n, producing the composite verged men-point rule for weighted integrls 743) where the reminder is bounded by n wt)ft) dt h i fx i ) + Rf, ξ, n) 744) Rf, ξ, n) f n h i σ i

339 33 J Roumeliotis n Error ) Error ) Error 3) Error rtio 3) Bound rtio 3) 4 970) 380) 480) 8 34-) 93-) 35-) ) 568-) 6-) ) 3-) 434-3) ) 30-3) 934-4) ) 794-4) 3-4) ) 98-4) 55-5) Tble 7 The error in evluting 745) under different qudrture rules The prmeter n is the number of smple points Proof The proof follows tht of Corollry 7 where it is observed tht the minimum bound 740) will occur t x i µ i Differentiting the right hnd side of 740) gives n d xj µ dξ j ) + σ j hj wξ i )x i x i ) ξ i x ) i + x i i j0 Inspection of the second derivtive t the root revels tht the sttionry point is minimum nd hence the result is proved 734 Numericl Results In this section, for illustrtion, the qudrture rule of subsection 733 is used on the integrl 745) 0 00t ln/t) cos4πt) dt Eqution 745) is evluted using the following three rules: ) the composite mid-point rule, where the grid hs uniform step-size nd the node is simply the mid-point of ech sub-intervl, ) the composite generlized mid-point rule 739) The grid, I n, is uniform nd the nodes re the men point of ech sub-intervl 74), 3) eqution 743) where the grid is distributed ccording to 74) nd the nodes re the sub-intervl mens 74) Tble 7 shows the numericl error of ech method for n incresing number of smple points For uniform grid, it cn be seen tht chnging the loction of the smpling point from the midpoint method ) to the men point method ) roughly doubles the ccurcy Chnging the grid distribution s well s the node point method 3) from the composite mid-point rule method ) increses the ccurcy by pproximtely n order of mgnitude It is importnt to note tht the nodes nd weights for method 3) cn be esily clculted numericlly using n itertive scheme For exmple on Pentium-III 550 MHz) personl computer, with n 64, clculting 74) nd 74) took close to 4 seconds

340 7 PRODUCT INEQUALITIES AND WEIGHTED QUADRATURE 33 Note tht equtions 74) nd 74) re quite generl in nture nd only rely on the weight insofr s knowledge of the first two moments is required This contrsts with Gussin qudrture where for n n point rule, the first n moments re needed or equivlently the n + coefficients of the continued frction expnsion 4, 4) to construct the pproprite orthogonl polynomil nd then rootfinding procedure is clled to find the bscisse This procedure, of course, cn be gretly simplified for the more well known weight functions 4 The second lst column of Tble 7 shows the rtio of the numericl errors for method 3) nd the lst column the rtio of the theoreticl error bound 743) 746) Bound rtio 3) Rf, ξ, n/) Rf, ξ, n) As n increses the numericl rtio pproches the theoreticl one The theoreticl rtio is consistently close to 4 This vlue suggests n symptotic form of the error bound ) 747) Rf, ξ, n) O n for the log weight Similr results hve been obtined for the other weights of subsection 73 This is consistent with mid-point type rules nd it is nticipted tht developing other product rules, for exmple generlized trpezoidl or Simpson s rule, will yield more ccurte results This is pursued in the next section where generlized trpezoid qudrture rule is developed 74 Weighted Boundry Point Lobtto) Integrl Inequlities In the previous section product integrl inequlities nd weighted qudrture rules were developed where smpling occurred t interior points In this section we develop nlogous results where the mpping f is smpled t the boundry points The inequlities presented here, which re vlid for mppings whose second derivtive exist, will be used to develop product trpezoidl-like qudrture rules In ddition to qudrture, these integrl inequlities hve been pplied to the numericl nlysis of first kind integrl equtions with symmetric kernels 38, Chpter 3, 37 We begin by defining weighted trpezoidl-type Peno kernel Lemm 75 Let x, b be fixed nd w be s given in Definition, then the Peno-type kernel, K, ), given by { t mt, x) µt, x) t ), t x 748) Kx, t) t u)wu) du mx, t) t µx, t) ), t > x is convex nd non-negtive for x b x

341 333 J Roumeliotis 749) Proof Differentition of 748) gives d Kx, t) dt t x wu) du < 0, if t < x, 0, if t x, > 0, if x < t b Eqution 749) immeditely revels the convexity of K In ddition, note tht K vnishes t its minimum t x, nd this property, with 749) suffices to prove tht K is non-negtive Theorem 76 Let f be n bsolutely continuous mpping defined on the finite intervl, b whose second derivtive exists nd let w be positive weight function s given in Definition Then, for fixed x, b the following product-trpezoidl like inequlities hold 750) wt)ft) dt m, x) f)+µ, x) )f ) ) mx, b) fb)+µx, b) b)f b) ) f { m, x) µ, x) ) + σ, x) f { + mx, b) b µx, b)) + σ x, b) }, f L, b m, x) µ, x) +mx, b) b µx, b) } + m, b)µ, b) m, x) bmx, b), f L, b The bound in 750) is minimized t the point x + b)/ nd the bound in 750) is minimized t the point x stisfying 75) m, x ) µ, x ) ) mx, b) b µx, b) ) Proof Integrting I Kx, t)f t) dt twice by prts gives 75) I wt)ft) dt m, x)f) m, x)µ, x) )f ) Using the well known properties of definite integrls 753) 754) I Kx, t)f t) dt f Kx, t) dt { f x t ) wt) dt + mx, b)fb) + b µx, b))f b) x t b) wt) dt The lst line being obtined by reversing the order of integrtion Mking use of eqution 75) nd substituting equtions 75)-77) into 754) returns 750) }

342 7 PRODUCT INEQUALITIES AND WEIGHTED QUADRATURE 334 mx{ Kx, ), Kxb, )} K x,b) x* K x,) b x Figure 7 A schemtic of the function mx{kx, ), Kx, b)} for x, b showing the minimum point x To find the vlue of x which minimizes 754), observe tht d dx Kx, t) dt x ) wx) x b) wx) wx)b ) x + b ) Since w is positive on, b), minimum occurs t the mid-point To prove 750), we mke use of 753) nd Lemm 75 Kx, t)f t) dt sup Kx, t) f t,b) mx{kx, ), Kx, b)} f f Kx, ) + Kx, b) + K, x) Kb, x) Simplifying the expression bove produces the desired result To obtin the optiml point x note tht Kx, ) is incresing nd Kx, b) is decresing with respect to x Thus the minimum of mx{kx, ), Kx, b)} will occur t Kx, ) Kx, b) s given in eqution 75) A schemtic is shown in Figure 7 The product inequlity my be useful s the bsis for composite weighted qudrture rule since only the first two moments re required for the rule For the bound in 750), knowledge of the third moment is needed The following corollries simplify nd in some cses consequently degrde) eqution 750), in tht the bound is given in terms of fewer moments The dditionl constrint tht b remin finite is lso imposed Corollry 77 Define Q to be Qx) m, x)f) m, x)µ, x) )f ) mx, b)fb) + b µx, b))f b)

343 335 J Roumeliotis Given the conditions in Theorem 76, the following inequlities hold 755) wt)ft) dt Qx) nd 756) f { } m, x)x )µ, x) ) + mx, b)b x)b µx, b)) wt)ft) dt Qx) f m, b) x + b + b ) We remrk tht the bound in 755) requires the first two moments of w, which contrsts with tht of 756) where only the first moment is needed In ddition, identifying the optiml point for the bound in 756) is obvious Proof To prove 755), we ppel to eqution 73) Applying 73) over the intervls, x nd x, b nd simplifying produces 755) To show tht 756) is true, note from 754) tht x t ) wt) dt + x t b) wt) dt m, b) mx { x ), x b) } m, b){ x) + x b) + x ) x b) } m, b) x + b + b ) Corollry 78 If w is bounded s well s integrble, then the following inequlity holds 757) b wt)ft) dt Qx) f w b ) x + b ) b ) + We remrk tht the bound in 757) does not require ny moments of w but is less ccurte in the sense tht it is n upper-bound for 750) x Proof The proof of eqution 757) follows gin from eqution 754) t ) wt) dt + x t b) wt) dt 3 w { x ) 3 x b) 3} w b ) x + b ) + b )

344 7 PRODUCT INEQUALITIES AND WEIGHTED QUADRATURE Development of Product-Trpezoidl Like Qudrture Rule Eqution 750) will be used s the bsis for product trpezoidl-like qudrture rule From 757), we cn see tht the error in pproximting wt)ft) dt is Ob ) 3 ) ssuming finite b ) Thus, to obtin n ccurte estimte to the definite integrl we require composite rule where the intervl, b is subdivided Usully this subdivision is uniform, but in this cse the grid should ccount for the presence of the weight In the next theorem, the intervl, b is divided into two optiml prts:, ξ, ξ, b This result is employed in the development of weighted qudrture rule Theorem 79 Two intervls) Let the conditions in Theorem 76 hold following two intervl product integrl inequlity holds 758) wt)ft) dt m, x ) f) + µ, x ) )f ) ) mx, x ) fξ) + µx, x ) ξ)f ξ) ) mx, b) fb) + µx, b) b)f b) ) f for x ξ x b The bound is minimized t the points { m, x ) µ, x ) ) + σ, x ) + mx, x ) µx, x ) ξ) + σ x, x ) + mx, b) b µx, b)) + σ x, b) }, 759) x + ξ, ξ µx, x ) nd x ξ + b Proof Define the kernel 760) Kx, x, ξ, t) { t x t u)wu) du, t, x < ξ t x t u)wu) du, ξ t, x b Employing the sme techniques s those in Theorem 76, nmely integrting Kx, x, ξ, t)f t) dt by prts twice nd simplifying, gives 76) wt)ft) dt m, x ) f) + µ, x ) )f ) ) mx, x ) fξ) + µx, x) ξ)f ξ) ) mx, b) fb) + µx, b) b)f b) ) f Kx, x, ξ, t)f t) dt { x t ) wt) dt + x x t ξ) wt) dt + Mking use of 75), 76) nd simplifying produces 758) } t b) wt) dt x The

345 337 J Roumeliotis Differentition revels the upper bound minimum If we let I b x t ) wt) dt + x x t ξ) wt) dt + x t b) wt) dt, then observe tht di x b dξ ξ t)wt) dt mx, x ) ξ µx, x ) ), x di b dx wx )ξ ) x + ξ ) nd di b dx wx )b ξ) x ξ + b ) Hence, the proof is complete Remrk 76 Substituting x nd x b in 758) produces the interior point inequlity 7) Theorem 70 Product trpezoidl-like qudrture rule) Define grid I n : ξ 0 < ξ < < ξ n < ξ n b on the intervl, b The following qudrture formul for weighted integrls is obtined 76) n wt)ft) dt h i fξ i )+h 0 µ 0 ξ 0 )f ξ 0 ) h n ξ n µ n )f ξ n )+Rf, w, ξ, x), where 763) Rf, w, ξ, x) n f h i σ i + h 0 µ 0 ξ 0 ) + h n ξ n µ n ), 764) nd 765) x i ξ i + ξ i+, i 0,, n, ξ i µx i, x i ), i,, n h i mx i, x i ), σ i σ x i, x i ), i,, n, h 0 mξ 0, x 0 ), h n mx n, ξ n ), µ 0 µξ 0, x 0 ), µ n µx n, ξ n ), σ 0 σ ξ 0, x 0 ), σ n σ x n, ξ n ) Proof For n n + point rule, we extend the kernel 760) to n brnches with ech brnch defined over ξ i, ξ i+, i 0,,, n Following the proof of Theorem 79 produces the desired qudrture rule The grid equtions 764) re simple to solve Any size grid cn be evluted very quickly, by using n itertive pproch, so long s the zero-th nd first moment of the weight re known In Figure 7 we show the node point distribution for logrithmic weight The clustering ner the origin is obvious nd is due to the presence of the singulrity t this point

346 7 PRODUCT INEQUALITIES AND WEIGHTED QUADRATURE 338 Figure 7 Node point distribution produced by solving eqution 764) for 3 point log rule over the unit intervl wt) ln/t)) Error n Eqution 76) Rtio) Atkinson, p 30 Error 3) from Tble ) 7480) 480) ) ) 703-) 35-) 6 08-) 35) 7-) 6-) ) 499) 78-) 434-3) ) 444) 659-3) 934-4) ) 409) 6-3) 3-4) ) 44) 399-4) 55-4) Tble 7 The error in evluting 745) using different qudrture rules n is the number of smple points 74 Numericl Experiment In Tble 7, we compre 76) with the interior point rule 739) of the previous section nd product trpezoidl type rule developed by Atkinson, p 30 We cn see tht the rule developed here compres fvourbly with others of similr order The rtio shows tht the rule is of order h 743 Some Prticulr Weighted Integrl Inequlities The results of Theorem 79 re tbulted for some of the more populr weight functions In ech cse x, ξ nd x re given by 759) so tht the bound is minimized 766) 743 Uniform Legendre) ft) dt b Logrithm b ) f) f ) b 3 f + b ) 3 ) + b b 4 fb) f b) 96 f b ) 3 767) ft) ln/t) dt f0) f 0) f ) f) f ) f

347 339 J Roumeliotis 7433 Jcobi 768) ft) dt f0) f 0) f45755) 0 t f) f ) f 769) 7434 Chebyshev ft) dt π t 3 f ) π 3 3 f ) π 6 3 f0) π 3 f) + π 3 3 f ) 6 f 7π 3) 7435 Lguerre The Lguerre weight is not immeditely pplicble since b is infinite nd hence by 759) so is ξ To produce Lguerre inequlity we ssume b is finite nd the tke the limit s b In this limit we obtin 770) e t ft) dt e )f0) e )f 0) e f) e ) f, 0 ssuming tht f o e x/) s x nd f is bounded on 0, ) 75 Weighted Three Point Integrl Inequlities In the previous two sections, interior point nd boundry point inequlties hve been investigted In this section we generlize nd combine the previous results vi prmeteriztion The prmeter distinguishes rule type nd t its extremes will produce n interior or boundry point inequlity The inequlities re clled three point rules At the end of this section three point rules re constructed by Grüss inequlity Three point qudrture rules of Newton-Cotes type hve been exmined in Cerone nd Drgomir 7 in which the error involved the behviour of, t most, first derivtive Riemnn nd Riemnn-Stieltjes integrls were exmined In the current section, weighted three point rules re investigted in which the error relies on the behviour of the first derivtive After developing the results in the initil subsections, composite qudrture rules re implemented nd results for log weight function re given nd compred with product-trpezoidl rule of Atkinson Theorem 7 ) Let f :, b R be differentible mpping on, b) whose derivtive is bounded on, b) nd denote f sup t,b) f t) < Further, let non-negtive weight function w ) hve the properties s outlined in Definition

348 7 PRODUCT INEQUALITIES AND WEIGHTED QUADRATURE 340 Then for x, b, α, x, β x, b, the following inequlity holds 77) wt)ft) dt m α, β) f x) + m, α) f ) + m β, b) f b) I α, x, β) f, where 77) I α, x, β) t, t, α k x, t) w t) dt nd k x, t) x t, t α, β b t, t β, b Proof Define the mpping K, ) :, b R by { m α, t), t, x 773) K x, t) m β, t), t x, b, where m, b) is the zeroth moment of w ) over the intervl, b nd is given by 75) It should be noted tht m c, d) will be non-negtive for d c Integrtion by prts gives, on using 773), K x, t) f t) dt producing the identity 774) x m α, t) f t) dt + m α, t) f t) x t x + m β, t) f t) m β, t) f t) dt b tx K x, t) f t) dt m α, β) f x)+m, α) f )+m β, b) f b) vlid for ll x, b w t) f t) dt, w t) f t) dt, Tking the modulus of 774) gives 775) w t) f t) dt m α, β) f x) + m, α) f ) + m β, b) f b) K x, t) f t) dt f K x, t) dt Now, we wish to determine K x, t) dt To this end notice tht, from 773), K x, t) is monotoniclly non-decresing function of t over ech of its brnches Thus, there re points α, x nd β x, b such tht K x, α) K x, β) 0 Thus, 776) K x, t) dt α m α, t) dt+ x α β m α, t) dt m β, t) dt+ x β m β, t) dt

349 34 J Roumeliotis Integrtion by prts gives, for exmple, α m α, t) dt t ) m α, t) α α + t t ) w t) dt α t ) w t) dt A similr development for the reminder of the three integrls on the right hnd side of 776) produces the result 777) K x, t) dt I α, x, β), where I α, x, β) is s given by 77) Combining 775) nd 777) produces the result 77) nd hence the theorem is proved It should be noted t this stge tht tking w ) reproduces the results of Cerone nd Drgomir 7 If α nd β b then weighted interior point rule is obtined If α β x, then weighted rule results where the function is evluted t the boundry points For α or β b then Rdu type rules re obtined while the current work will focus on Lobtto type rules llowing smpling t both ends of the boundry Corollry 7 Inequlity 77) is minimized t x x where x stisfies 778) m α, x ) m x, β ), α + x nd Proof From 77) - 77), I α, x, β) my be written s 779) I α, x, β) α t ) w t) dt + + x α β x x t) w t) dt t x) w t) dt + β x + b β b t) w t) dt, where α, x nd β x, b Eqution 779) could equivlently be written in terms of its zeroth nd first moments s given by 75) Differentiting 779) with respect to α, β nd x gives 780) where I α A α, x) w x), I B β, x) w x) nd I β 78) A α, x) α + x), B β, x) β x + b) m α, x) m x, β), x nd m, ) is defined by 75) An inspection of the second derivtives demonstrtes tht 779) is convex on using the fct tht w t) is non-negtive for t, b) Thus, I is miniml t the zeros of 780) nd so the corollry is proven Corollry 7 investigtes the problem of determining the optiml choice of α, x nd β tht produce the tightest bound The following corollry gives corser bounds lthough the bound my be esier to implement

350 7 PRODUCT INEQUALITIES AND WEIGHTED QUADRATURE 34 Corollry 73 Let the conditions be s in Theorem 7 Then the following inequlities hold 78) w t) f t) dt m α, β) f x) + m, α) f ) + m β, b) f b) w K x) f, w K x) where 783) K x) b ) + x + b ) + α + x ) + β x + b ) nd 784) K x) b + α + x + β x + b + x + b + α + x β x + b with g : g s) ds mening g L, b, the liner spce of bsolutely integrble functions nd g : sup t,b g t) < Proof From Theorem 7 nd equtions 77) - 77), 773) nd 777) we hve Now, I α, x, β) where k x, t) is s defined in 77) K x, t) w t) dt k x, t) w t) dt w k x, t) dt k x, t) w t) dt w sup t,b k x, t) Some stright forwrd evlution gives k x, t) dt α ) + x α) + β x) + b β), which my redily be shown to equl K x) s given by 783) through using the identity X + Y ) ) X + Y X Y + three times Further, sup k x, t) mx {α, x α, β x, b β}, t,b which cn be shown to equl K x) s given by 784) from using the result 785) mx {X, Y } X + Y + X Y,,

351 343 J Roumeliotis three times Hence the corollry is proven It should be noted tht the tightest bounds re obtined t x +b nd α +x, β x+b Tht is, t their respective midpoints The optiml smpling scheme is independent of the weight Theorem 74 Let f : I R R be differentible mpping on I the interior of I) nd, b I re such tht b > If f L, b, then f f t) dt < In ddition, let non-negtive weight function w ) hve the properties s outlined in Definition Then for x, b, α, x nd β x, b the following inequlity holds 786) w t) f t) dt m α, β) f x) + m, α) f ) + m β, b) f b) where θ α, x, β) f, 787) θ α, x, β) 4{ m, b) + m α, x) m, α) + m β, b) m x, β) + m, x) m x, b) + m α, x) m, α) m β, b) m x, β) } nd m, b) is the zeroth moment of w ) over, b s defined by 75) Proof From identity 774) we obtin, from tking the modulus θ α, x, β) sup K x, t), t,b where K x, t) is s given by 773) As discussed in the proof of Theorem 7, K x, t) is monotonic non-decresing function of t in ech of its two brnches so tht θ α, x, β) mx {m, α), m α, x), m x, β), m β, b)} Now, using eqution 785) we hve m mx {m, α), m α, x)} m, x) + m α, x) m, α) nd m mx {m x, β), m β, b)} m x, b) + m β, b) m x, β), to give θ α, x, β) mx {m, m } m + m + m m nd hence the result 787) is obtined fter some simplifiction nd the theorem is proved Remrk 77 It should be noted tht the tightest bound in 787) is obtined when α, x nd β re tken s their respective medins Thus, the best qudrture rule

352 7 PRODUCT INEQUALITIES AND WEIGHTED QUADRATURE 344 in the bove sense is given by 788) w t) f t) dt m, α) f ) + m α, β)f m, b) x) + m β, b)f b) f 4, where m, x) m x, b), m, α) m α, x) nd m β, b) m x, β) 75 Development of Qudrture Rule The following theorem will be useful in determining the prtition for composite qudrture rules The optiml prtition in terms of the prtition tht provides the tightest bounds will be determined The optiml qudrture rules will result for f L, b If f L, b similr development my be followed but will not be pursued further here Theorem 75 Let the conditions of Theorem 7 hold nd let ξ prtition the intervl, b into two Then the following inequlity holds 789) w t) f t) dt m, α ) f ) + m α, β ) f x ) + m β, α ) f ξ) + m α, β ) f x ) + m β, b) f b) J z, ξ) f, where 790) J z, ξ) J z, ξ) + J z, ξ) with 79) nd 79) J z, ξ) k x, t) ξ z T i α i, x i, β i ), i,, z z z, k x, t) w t) dt, J z, ξ) t, t, α x t, t α, β ξ t, t β, ξ, k x, t) Further, α x β ξ nd ξ α x β b ξ k x, t) w t) dt t ξ, t ξ, α x t, t α, β b t, t β, b Proof The proof follows tht of Theorem 7 A subscript of is used to denote prmeters in the intervl, ξ nd for prmeters in ξ, b Integrtion by prts of ξ K x, t) f t) dt produces n identity similr to 774) with b replced by ξ nd x by x Similrly for ξ K x, t) f t) dt produces n identity like 774) with replced by ξ nd x by x Summing the two results produces n identity over, b Tking the modulus nd using the tringle inequlity, relying hevily on 77) gives the stted result fter collecting the terms in order Here on, ξ, α, x, β, b) re replced by α, x, β, ξ) nd on ξ, b,, α, x, β) re replced by ξ, α, x, β ) Hence the theorem is proved

353 345 J Roumeliotis Corollry 76 The optiml loction of the prmeters in Theorem 75 re α α +x, β β x +ξ, α α ξ +x, β β x +b nd x, x nd ξ stisfy the following respective equtions m α, x ) m x, β ), m α, x ) m x, β ) nd m β, ξ ) m ξ, α ) Proof The proof of this corollry closely follows tht of Corollry 7 From 790) - 79), differentition of J with respect to α, x, β, ξ, α, x, β ) produces, on equting to zero, seven simultneous equtions Using the fct tht the weight function is ssumed to be positive, then the solution of the seven simultneous equtions give the point t which n optiml bound is produced, since n inspection of the second derivtives redily demonstrtes the convexity of the function J The results in Theorem 75 my be used to develop composite qudrture rule To this end, define grid I n : ξ 0 < ξ < < ξ n < ξ n b on the intervl, b, with x i ξ i, ξ i+ for i 0,,, n The following qudrture formul for weighted integrls is obtined which relies only on the first two moments of the weight function Theorem 77 Let the conditions in Theorem 75 hold, then following weighted qudrture rule holds 793) where 794) Af, ξ, x) nd n i wt)ft) dt Af, ξ, x) + Rf, ξ, x) n mα i, β i )fx i )+mξ 0, α )fξ 0 )+ mβ i, ξ i )fξ i )+mβ n, ξ n )fξ n ) 795) Rf, ξ, x) f Mξ 0, ξ n ) The prmeters x i, α i, β i nd ξ i stisfy i n Mα i, β i ) + Mβ i, ξ i ) i ) + ξ n mβ n, ξ n ) ξ 0 mξ 0, α ) 796) mα i, x i ) mx i, β i ), α i ξ i + x i, β i x i + ξ i for i,,, n, nd 797) mβ i, ξ i ) mξ i, α i+ ), for i,,, n Proof Using the results of Theorems 7 nd 75 over ξ i, ξ i+ for i 0,,, n nd summing redily produces the result fter using Corollries 7 nd 76 to simplify

354 7 PRODUCT INEQUALITIES AND WEIGHTED QUADRATURE Numericl Results In this section we illustrte the ppliction of the composite qudrture rule developed in the previous section to pproximte the integrls 798) 0 ln/t) t + dt nd 0 e /t ln/t) dt The integrls re evluted using the composite rule 793) nd the product-trpezoidl s described in, p 30 The first integrl, 798), hs been used to demonstrte the product-trpezoidl nd s result we cn compre the performnce with the rule developed here Note tht 793) is first-order rule in tht it ws derived for the clss of once-differentible functions This contrsts with the producttrpezoidl rule which is of second order Thus, to investigte the effects of rule order, we lso pply these rules to 798) In contrst with 798), the integrnd of 798) increses with the order of its derivtive Tble 73 shows the numericl error in evluting 798) using 793) for n incresing number of intervls We note tht the nodes nd weights of the qudrture rule re obtined by solving the 4n simultneous equtions 796) nd 797) It is simple mtter to implement numericl procedure to solve these equtions itertively with n initil uniform mesh For exmple on Pentium-90 personl computer, with n 3, clculting 796) nd 797) to 4 digit ccurcy took close to 4 seconds Inspection of Tble 73 revels tht more ccurte result is obtined for 798) thn for 798) This is probbly due to the nture of the integrnds The theoreticl error rtio is consistently close to This vlue confirms tht, due to its development, the qudrture rule is t lest of first order The numericl error rtios re somewht lrger, these vlues suggest n symptotic form of the error bound 799) Rf, ξ, x) O ) n γ, where γ n Eqution 798) Eqution 798) Theoreticl Reltive Error Error Rtio Reltive Error Error Rtio Error Rtio 64-) 77-) ) ) ) ) ) ) ) ) ) ) Tble 73 The reltive error in evluting 798) using 793), where n is the number of intervls In Tble 74 the errors in employing the product-trpezoidl rule re presented The error rtios re consistently close to 4 which simply reflects the fct tht the rule is of second order This rule ws developed by employing liner pproximtion for the weighted integrnd - higher order pproximtion thn tht used here This

355 347 J Roumeliotis rule performs better thn 793) for 798) since the integrnd is well behved nd its mgnitude decreses s its derivtives increse In contrst, the producttrpezoidl rule is inferior to 793) for 798) This integrnd is not well behved nd its integrl is better suited to 793) which ws developed for more generl clss of function n Eqution 798) Eqution 798) Reltive Error Error Rtio Reltive Error Error Rtio 7-3) 49-) ) ) ) ) ) ) ) ) ) ) 400 Tble 74 The reltive error in evluting 798) using the product trpezoidl rule, p 30, where n is the number of intervls We note tht the product-trpezoidl rule employs uniform mesh nd the behviour of the weight function, wt), is ccounted for in the qudrture rule weight Rules of this type were explored in Subsection 734, where one-point, second order product rule ws developed In this subsection, we showed tht, for the log weight, employing non-uniform mesh, similr to 797) increses ccurcy by fctor of more thn two for f L, b Finlly, we note tht the rule developed here is composite in nture nd identifies n optiml prtition for n rbitrry weight This contrsts with Guss qudrture 43 which is not composite nd hence provides no informtion s to how one should prtition 75 Appliction of Grüss Type Inequlities Grüss inequlity 5 provides bound for the difference between n integrl of product nd product of integrls, viz 700) b b fx)gx) dx b b ) fx) dx gx) dx Γ γ)φ φ), 4 where f, g re such tht the integrls bove exist nd γ fx) Γ, φ gx) Φ The inequlity 700) hs been used with much success with Peno-kernel inspired pplictions For exmple, the product integrnd of 70) is n idel cndidte for 700) In ddition, if one of the functions in 700) is explicitly known s is the cse in 70)) then bound my be improved 7, 4 To this end, define 70) T f, g) b fx)gx) dx b ) fx) dx gx) dx

356 7 PRODUCT INEQUALITIES AND WEIGHTED QUADRATURE 348 Using Korkine s identity, it hs been shown tht 70) T f, g) T / f, f)t / g, g), nd 703) 704) T f, f) Γ b 4 Γ γ) ) fx) dx b fx) dx γ ) Thus equtions 70) 704) provide us with the mens of writing down mny Grüss type inequlities; ech depending on the level to which the integrnds re known Theorem 78 Let f nd w be s defined previously nd let, b be finite intervl The following three point Grüss type inequlities for weighted integrls hold ) fb) f) wt)ft) dt m, b)fx) + m, b)x µ, b)) b 705) { b Γ γ) b K x, t) dt m, b)x µ, b)) b 706) ) / Γ γ)b ) m, b) m, x) x µ, b)) 707) m, b) x µ, b)) + mx, b) b b ) / } / 708) Γ γ)b )m, b), 4 where x b, γ f t) Γ nd K, ) is the kernel defined in 79) Proof The left hnd side of ech inequlity is simply T f ), Kx, ) ) Using the fct tht Kx, t) is bounded by mx, b) Kx, t) m, x), t, b, for fixed x nd Kx, t) dt m, b)x µ, b)) we cn bound T f ), Kx, ) ) by pplying 70) 704) The results re

357 349 J Roumeliotis 709) 70) 7) 7) T f ), Kx, ) ) Γ γ)t / Kx, ), Kx, ) ) Γ γ) { { b Γ γ)m, b) m, x) b Kx, t) dt + mx, b) Kx, t) dt } / } / Simplifying redily produces 70) 705) It is of interest to compre the left hnd side of the inequlities in Theorems 77 nd 78 The derivtive f x) in Theorem 77 is replced with the secnt slope fb) f) b in Theorem 78 Corollry 79 The following men point weighted integrl inequlity holds 73) { b wt)ft) dt m, b)fµ, b)) b Γ γ) b K µ, b), t) dt } / 74) 75) Γ γ)b ) m, µ, b))mµ, b), b) Γ γ)b )m, b) 4 Proof Substitute x µ, b) into the inequlities of Theorem 78 The men x µ, b) gretly simplifies the inequlities of Theorem 78, but this point does not necessrily minimise ny of the upper bounds This is done in the following subsection for prticulr weights 753 Grüss-type Inequlities for Some Weight Functions The inequlities in Theorem 78 become more corse s they become simpler to evlute Even so, 705) cn be evluted for mny of the populr weight functions In the following we tbulte the inequlity 705) for the some weight functions The minimum point is lso identified 753 Legendre Substituting wt) into 705) gives 76) ft) dt b )fx) + fb) f) ) x + b ) 4 3 Γ γ)b ) 0 The bove inequlity is vlid for ll x, b

358 7 PRODUCT INEQUALITIES AND WEIGHTED QUADRATURE Logrithm Substituting wt) ln/t), 0, b gives 77) ft) ln/t) dt fx) + f) f0) ) x 4) 0 7 Γ γ) x + 648x 96x lnx) The bound is minimized t x Jcobi With the Jcobi weight, we hve 78) ft) dt fx) + f) f0) ) x ) t 3 6 Γ γ) x 36x + 48x 3/ 0 The bound is minimized t x Chebyshev The Chebyshev weight gives 79) ft) dt πfx) π ) f) f ) x t Γ γ) 8πx rcsinx) + 8π x 4 6 x π The bound is minimized t the boundry points x ±

359 Bibliogrphy GA ANASTASSIOU, Ostrowski type inequlities, Proc Amer Mth Soc, 3) 995), KE ATKINSON, An Introduction to Numericl Anlysis, John Wiley nd Sons, New York, NS BARNETT, P CERONE, SS DRAGOMIR nd J ROUMELIOTIS, Some inequlities for the expecttion nd vrince of rndom vrible whose pdf is n-time differentible, J Inequl Pure Appl Mth, ) 000), Article 4 NS BARNETT, IS GOMM nd L ARMOUR, Loction of the optiml smpling point for the qulity ssessment of continuous strems, Austrl J Sttist, 37) 995), JR BLAKE nd DC GIBSON, Cvittion bubbles ner boundries, Ann Rev Fluid Mech, 9 987), CH BREBBIA nd J DOMINGUEZ, Boundry elements: n introductory course, Computtionl Mechnics, Southmpton, P CERONE nd SS DRAGOMIR, Three point qudrture rules involving, t most, first derivtive, submitted to SIAM J Mth Anl, 999) 8 P CERONE, SS DRAGOMIR nd J ROUMELIOTIS, An inequlity of Ostrowski type for mppings whose second derivtives belong to L p, b) nd pplictions, Accepted for publiction in The Journl of the Indin Mthemticl Society, 998) 9, An inequlity of Ostrowski type for mppings whose second derivtives re bounded with pplictions, Est Asin Mth J, 5) 999), 9 0, An inequlity of Ostrowski type for mppings whose second derivtives belong to L, b) nd pplictions, Honm Mth J, ) 999), 7 37, Some Ostrowski type inequlities for n-time differentible mppings nd pplictions, Demonstrtio Mth, 34) 999), P CERONE, J ROUMELIOTIS nd G HANNA, On weighted three point qudrture rules, ANZIAM J, 4E) 000), C340 C36 ONLINE Avilble online from 3 Lj DEDIĆ, M MATIĆ nd J PEČARIĆ, On some generliztions of Ostrowski inequlity for Lipschitz functions nd functions of bounded vrition, Mth Inequl Appl, 3) 000), 4 4 SS DRAGOMIR, Better bounds in some Ostrowski-Grüss type inequlities, Accepted for publiction in Tmkng J Mth, ONLINE Preprint vilble; RGMIA Res Report Coll, 3) Article3 5, The Ostrowski integrl inequlity for mppings of bounded vrition, Bull Austrl Mth Soc, ), , Ostrowski s inequlity for monotonous mppings nd pplictions, J KSIAM, 3) 999), , The Ostrowski s integrl inequlity for Lipschitzin mppings nd pplictions, Comput Mth Appl, ), SS DRAGOMIR, P CERONE nd J ROUMELIOTIS, A new generliztion of Ostrowski s integrl inequlity for mppings whose derivtives re bounded nd pplictions in numericl integrtion nd for specil mens, Appl Mth Lett, 3 000), SS DRAGOMIR, P CERONE, J ROUMELIOTIS nd S WANG, A weighted version of Ostrowski inequlity for mppings of Hölder type nd pplictions in numericl nlysis, Bull Mth Soc Sc Mth Roumnie, 490)4) 999),

360 BIBLIOGRAPHY 35 0 SS DRAGOMIR nd S WANG, A new inequlity of Ostrowski s type in L norm nd pplictions to some specil mens nd to some numericl qudrture rules, Tmkng J Mth, 8 997), SS DRAGOMIR, P CERONE nd J ROUMELIOTIS, A new generliztion of Ostrowski s integrl inequlity for mppings whose derivtives re bounded nd pplictions in numericl integrtion nd for specil mens, Appl Mth Lett, 3 000), 9 5 H ENGELS, Numericl qudrture nd cubture, Acdemic Press, New York, AM FINK, Bounds on the devition of function from its verges, Czechoslovk Mth J, 4 99), W GAUTSCHI, Algorithm 76: ORTHPOL - pckge of routines for generting orthogonl polynomils nd Guss-type qudrture rules, ACM Trns Mth Softwre, 0 994), 6 5 G GRÜSS, Über ds mximum des bsoluten betrges von b b fx)gx) dx b b ) fx) dx gx) dx, Mth Z, ), AR KROMMER nd CW UEBERHUBER, Numericl integrtion on dvnced computer systems, Lecture Notes in Computer Science, 848, Springer-Verlg, Berlin, M MATIĆ, JE PEČARIĆ nd N UJEVIĆ, On new estimtion of the reminder in generlized Tylor s formul, Mth Inequl Appl, 3) 999), GV MILOVANOVIĆ, On some integrl inequlities, Univ Beogrd Publ Elektrotehn Fk Ser Mt, No ), GV MILOVANOVIĆ nd JE PEČARIĆ, On generliztions of the inequlity of A Ostrowski nd some relted pplictions, Univ Beogrd Publ Elektrotehn Fk Ser Mt, No ), GV MILOVANOVIĆ nd I Z MILOVANOVIĆ, On generliztion of certin results of A Ostrowski nd A Lupş, Univ Beogrd Publ Elektrotehn Fk Ser Mt, No ), DS MITRINOVIĆ, JE PEČARIĆ nd AM FINK, Inequlities for Functions nd their Integrls nd Derivtives, Kluwer Acdemic, Dordrecht, A OSTROWSKI, Über die bsolutbweichung einer differentiierbren funktion von ihrem integrlmittelwert, Comment Mth Helv, 0 938), TC PEACHEY, A McANDREW nd SS DRAGOMIR, The best constnt in n inequlity of Ostrowski type, Tmkng J Mth, 303) 999), 9 34 CEM PEARCE nd J PEČARIĆ, On Anstssiou s generliztions of the Ostrowski inequlity nd relted results, J Comput Anl Appl, 3) 000), J PEČARIĆ nd S VAROŠANEC, Hrmonic polynomils nd generliztions of Ostrowski inequlity with pplictions in numericl integrtion, submitted to WCNA JE PEČARIĆ nd B SAVIĆ, O novom postupku rzvijnj funkcij u red i nekim primjenm, Zb Rd, Beogr) 9 983), J ROUMELIOTIS, Product trpezoidl-like inequlities with pplictions in numericl integrtion nd integrl equtions, mnuscript in preprtion for submission to J Inequl Pure Appl Mth 38, A boundry integrl method pplied to Stokes flow, PhD thesis, The University of New South Wles, 000, ONLINE A PDF version is vilble from 39 J ROUMELIOTIS, P CERONE nd SS DRAGOMIR, An Ostrowski type inequlity for weighted mppings with bounded second derivtives, J KSIAM, 3) 999), J ROUMELIOTIS nd GR FULFORD, Droplet interctions in creeping flow, Comp nd Fluids, 94) 000), H RUTISHAUSER, Algorithm 5: WEIGHTCOEFF, CACM, 50) 96), , On modifiction of the QD-lgorithm with Greffe-type convergence, Proceedings of the IFIPS Congress, Munich), AH STROUD nd D SECREST, Gussin Qudrture Formuls, Prentice-Hll, Englewood Cliffs, NJ, 966

361 CHAPTER 8 Some Inequlities for Riemnn-Stieltjes Integrl by SS DRAGOMIR Abstrct In this chpter we present some recent results of the uthor concerning certin inequlities of Trpezoid type, Ostrowski type nd Grüss type for Riemnn- Stieltjes integrls nd their nturl ppliction to the problem of pproximting the Riemnn-Stieltjes integrl 8 Introduction Let f nd u denote rel-vlued functions defined on closed intervl, b of the rel line We shll suppose tht both f nd u re bounded on, b; this stnding hypothesis will not be repeted A prtition of, b is finite collection of nonoverlpping intervls whose union is, b Usully, we describe prtition I n by specifying finite set of rel numbers x 0, x,, x n ) such tht x 0 < x < < x n < x n b, nd the subintervls occurring in the prtition I n re the intervls x k, x k, k,, n Definition 4 If I n is prtition of, b, then the Riemnn-Stieltjes sum of f with respect to u, corresponding to I n x 0, x,, x n ) is rel number σ I n ; f, u) of the form n 8) σ I n ; f, u) f ξ k ) {u x k ) u x k )} k Here we hve selected numbers ξ k stisfying x k ξ k x k for k,,, n Definition 5 We sy tht f is integrble with respect to u on, b if there exists rel number I such tht for every number ε > 0 there is prtition I nε) of, b such tht, if I n is ny refinement of I nε) nd σ I n ; f, u) is ny Riemnn-Stieltjes sum corresponding to I n, then 8) σ I n ; f, u) I < ε 353

362 8 SOME INEQUALITIES FOR RIEMANN-STIELTJES INTEGRAL 354 In this cse the number I is uniquely determined nd is denoted by I f du f t) du t) ; it is clled the Riemnn-Stieltjes integrl of f with respect to u over, b We cll the function f the integrnd nd u the integrtor Sometimes we sy tht f is u integrble if f is integrble with respect to u For the fundmentl properties of Riemnn-Stieltjes integrls relted to: the Cuchy criterion for integrbility, the functionl properties of the integrl, the integrtion by prts formul, the modifiction of the integrl, the existence of the integrl, the evlution of the integrl first men vlue theorem nd second men vlue theorem) nd other properties, we refer the reder to the clssicl book 5, by R G Brtle In this chpter we point out some recent results by the uthor concerning certin inequlities of Trpezoid type, Ostrowski type nd Grüss type for Riemnn-Stieltjes integrls in terms of certin Riemnn-Stieltjes sums, generlised mid-point sums, generlised trpezoidl sums, etc For comprehensive study of Newton-Cotes qudrture formule for Riemnn- Stieltjes integrls nd their pplictions to numericl evlutions of life distributions, see the pper 73 by M Tortorell nd the references therein The chpter is structured s follows: The first section dels with the estimtion of the mgnitude of the difference f ) + f b) u b) u ) f t) du t), where f is of Hölder type nd u is of bounded vrition, nd vice vers The second section provides n error nlysis for the quntity f x) u b) u ) f t) du t), x, b, which is commonly known in the literture s n Ostrowski type inequlity, for the sme clsses of mppings Finlly, the lst section dels with Grüss type inequlities for the Riemnn-Stieltjes integrls, tht is, obtining bounds for the quntity f t) du t) u b) u ) b f t) dt All the sections contin implementtion for composite qudrture formule A lrge number of references to the recent ppers done by the Reserch Group in Mthemticl Inequlities nd Applictions RGMIA, re included see for exmple -4, 6-64 nd 7)

363 355 SS Drgomir 8 Some Trpezoid Like Inequlities for Riemnn-Stieltjes Integrl 8 Introduction The following inequlity is well known in the literture s the trpezoid inequlity : f ) + f b) b 83) b ) f t) dt b )3 f, where the mpping f :, b R is ssumed to be twice differentible on, b), with its second derivtive f :, b) R bounded on, b), tht is, f : sup t,b) f t) < The constnt is shrp in 83) in the sense tht it cnnot be replced by smller constnt If I n : x 0 < x < < x n < x n b is division of the intervl, b nd h i x i+ x i, ν h) : mx {h i i 0,, n }, then the following formul, which is clled the trpezoid qudrture formul 84) T f, I n ) n f x i ) + f x i+ ) pproximtes the integrl f t) dt with n error of pproximtion R T f, I n ) which stisfies the estimte 85) R T f, I n ) n f h 3 i b f ν h) In 85), the constnt is shrp s well If the second derivtive does not exist or f is unbounded on, b), then we cnnot pply 85) to obtin bound for the pproximtion error It is importnt, therefore, tht we consider the problem of estimting R T f, I n ) in terms of lower derivtives Define the following functionl ssocited to the trpezoid inequlity 86) Ψ f;, b) : f ) + f b) b ) where f :, b R is n integrble mpping on, b The following result is known 34 see lso 60 or 0): h i f t) dt Theorem 8 Let f :, b R be n bsolutely continuous mpping on, b Then b ) 4 f if f L, b ; b ) 87) Ψ f;, b) + q f p if f L p, b, p >, p + q ; q+) q b f,

364 8 SOME INEQUALITIES FOR RIEMANN-STIELTJES INTEGRAL 356 where p re the usul p norms, ie, nd respectively f : ess sup f t), t,b f p : f : f t) p dt ) p f t) dt,, p > The following corollry for composite formule holds 34 Corollry 8 Let f be s in Theorem 8 Then we hve the qudrture formul 88) f x) dx T f, I n ) + R T f, I n ), where T f, I n ) is the trpezoid rule nd the reminder R T f, I n ) stisfies the estimtion 4 f n h i if f L, b ; n ) 89) R T f, I n ) f p hq+ q i if f L p, b, q+) q f ν h) p >, p + q ; A more generl result concerning trpezoid inequlity for functions of bounded vrition hs been proved in the pper 36 see lso 0) Theorem 83 Let f :, b R be mpping of bounded vrition on, b nd denote b f) s its totl vrition on, b Then we hve the inequlity 80) Ψ f;, b) b b ) f) The constnt is shrp in the sense tht it cnnot be replced by smller constnt The following corollry which provides n upper bound for the pproximtion error in the trpezoid qudrture formul, for f of bounded vrition, holds 36 Corollry 84 Assume tht f :, b R is of bounded vrition on, b Then we hve the qudrture formul 88) where the reminder stisfies the estimte 8) R T f, I n ) b ν h) f) The constnt is shrp For other recent results on the trpezoid inequlity see the books 60 nd 70 where further references re given

365 357 SS Drgomir 8 A Trpezoid Formul for the Riemnn-Stieltjes Integrl The following theorem generlizing the clssicl trpezoid inequlity for mppings of bounded vrition holds 57 Theorem 85 Let f :, b K K R,C) be p H Hölder type mpping, tht is, it stisfies the condition 8) f x) f y) H x y p for ll x, y, b, where H > 0 nd p 0, re given, nd u :, b K is mpping of bounded vrition on, b Then we hve the inequlity: 83) Ψ f, u;, b) b p H b )p u), where Ψ f, u;, b) is the generlized trpezoid functionl 84) Ψ f, u;, b) : f ) + f b) u b) u )) f t) du t) The constnt C on the right hnd side of 83) cnnot be replced by smller constnt Proof It is well known tht if g :, b K is continuous nd v :, b K is of bounded vrition, then the Riemnn-Stieltjes integrl g t) dv t) exists nd the following inequlity holds: b 85) g t) dv t) sup g t) v) Using this property, we hve 86) t,b Ψ f, u;, b) ) f ) + f b) f t) du t) sup f ) + f b) b f t) u t) t,b As f is of p H Hölder type, then we hve f ) + f b) f t) f ) f t) + f b) f t) 87) f ) f t) + b) f t) H t )p + b t) p Now, consider the mpping γ t) t ) p + b t) p, t, b, p 0, Then γ t) p t ) p p b t) t 0 iff t +b nd γ t) > 0 on ), +b, γ t) < 0 on +b, b), which shows tht its mximum is relized t t +b nd mx t,b γ t) γ ) +b p b ) p

366 8 SOME INEQUALITIES FOR RIEMANN-STIELTJES INTEGRAL 358 Consequently, by 87), we hve ) sup f ) + f b) p f t) b H t,b Using 86), we obtin the desired inequlity 83) To prove the shrpness of the constnt, ssume tht 83) holds with constnt C > 0 Tht is 88) Ψ f, u;, b) C b p H b )p u) Choose f, u : 0, R, f x) x p, p 0, nd u x) x, x 0, We observe tht f is of p H Hölder type with H nd u is of bounded vrition, then, by 88) we cn obtin p + C, for ll p 0, p Tht is, C p p + p, for ll p 0, Letting p 0+, we get C nd the theorem is completely proved The following corollries re nturl consequences of 83): Corollry 86 Let f be s bove nd u :, b R be monotonic mpping on, b Then we hve 89) Ψ f, u;, b) p H b )p u b) u ) The proof is obvious by the bove theorem, tking into ccount the fct tht monotonic mppings re of bounded vrition nd for such functions u, we hve b u) u b) u ) Corollry 87 Let f be s bove nd u :, b K be Lipschitzin mpping with the constnt L > 0 Then 80) Ψ f, u;, b) p HL b )p+ The proof follows by Theorem 85, tking into ccount tht ny Lipschitzin mpping u is of bounded vrition nd b u) L b ) Corollry 88 Let f be s bove nd G :, b R be the cumultive distribution function of certin rndom vrible X Then f ) + f b) b 8) f t) dg t) p H b )p The proof is obvious by the bove theorem, tking into ccount the fct tht G b), G ) 0 nd b G)

367 359 SS Drgomir Remrk 8 If we ssume tht g :, b, b)) K is continuous, then u x) x g t) dt is differentible, u b) g t) dt, u ) 0, nd b u) g t) dt Consequently, by 83), we obtin f ) + f b) b 8) g t) dt f t) g t) dt p H b )p g t) dt From 8), we get weighted version of the trpezoid inequlity, f ) + f b) b 83) f t) g t) dt g t) dt p H b )p, provided tht g t) 0, t, b nd g t) dt 0 We give now some exmples of weighted trpezoid inequlities for some of the most populr weights Exmple Legendre) If g t), t, b, then we get the following trpezoid inequlity for Hölder type mppings : f ) + f b) 84) f t) dt b p H b )p Exmple Logrithm) If g t) ln ) t, t 0,, f is of p Hölder type on 0, nd the integrl 0 f t) ln ) t dt is finite, then, by 83), we obtin 85) f 0) + f ) Exmple 3 Jcobi) If g t) 0 0 f t) ln t ) dt p H t, t 0,, f is s bove nd the integrl ft) t dt is finite, then by 83), we obtin 86) f 0) + f ) f t) dt t p H Finlly, we hve the following: Exmple 4 Chebychev) If g t) t, t, ), f is of p Hölder type on, ) nd the integrl 87) ft) t f ) + f ) 0 dt is finite, then π f t) t dt H 83 Approximtion of the Riemnn-Stieltjes Integrl Consider the prtition I n : x 0 < x < < x n < x n b of the intervl, b, nd define h i : x i+ x i i 0,, n ), ν h) : mx {h i i 0,, n } nd the sum 88) T n f, u, I n ) : n f x i ) + f x i+ ) u x i+ ) u x i )

368 8 SOME INEQUALITIES FOR RIEMANN-STIELTJES INTEGRAL 360 The following pproximtion of the Riemnn-Stieltjes integrl in terms of T n f, u, I n ) holds 57 Theorem 89 Let f :, b K be p H Hölder type mpping on, b p 0, ) nd u :, b K be function of bounded vrition on, b Then 89) f t) du t) T n f, u, I n ) + R n f, u, I n ), where T n f, u, I n ) is the generlized trpezoidl formul given by 88), nd the reminder R f, u, I n ) stisfies the estimte 830) R n f, u, I n ) b p H ν h)p u) Proof We pply Theorem 85 on every subintervl x i, x i+ i 0,, n ) to obtin f x i ) + f x i+ ) xi+ 83) u x i+ ) u x i ) f t) du t) x i p Hhp i x i+ x i u) Summing the inequlities 83) over i from 0 to n nd using the generlized tringle inequlity, we obtin R f, u, I n, ξ) n f x i ) + f x i+ ) n p H h p i x i+ x i b p H ν h)p u), nd the bound 830) is proved xi+ u x i+ ) u x i ) x i u) H ν h)p p n x i+ x i u) f t) du t) The following corollries hve useful pplictions Corollry 80 Let f be s in Theorem 89 nd u :, b R be monotonic mpping on, b Then we hve the formul 89) nd the reminder term R n f, u, I n ) stisfies the estimte 83) R n f, u, I n ) p H ν h)p u b) u ) Using Corollry 87, we lso point out the following result

369 36 SS Drgomir Corollry 8 Let f be s in Theorem 89 nd u :, b K be Lipschitzin mpping with the constnt L > 0 Then we hve the formul 89), for which the reminder will stisfy the bound: 833) R n f, u, I n ) n p HL h p+ i p HL b ) ν h)p We now point out some qudrture formule of trpezoid type for weighted integrls Let us ssume tht g :, b K is continuous nd f :, b K is of r H Hölder type on, b For given prtition I n of the intervl, b, consider the sum 834) T n f, g, I n ) : n We cn stte the following corollry f x i ) + f x i+ ) xi+ x i g s) ds Corollry 8 Let f :, b K be of r H Hölder type nd g :, b K be continuous on, b Then we hve the formul 835) g t) f t) dt T n f, g, I n ) + R n f, g, I n ), where the reminder term R n f, g, I n ) stisfies the estimte 836) R n f, g, I n ) p H ν h)p g s) ds Proof Apply the inequlity 8) on the intervls x i, x i+ i 0,, n ) to obtin f x i ) + f x i+ ) xi+ xi+ g s) ds f t) g t) dt x i x i xi+ p Hhp i g s) ds x i Summing over i from 0 to n nd using the generlized tringle inequlity, we cn stte R n f, g, I n ) n f x i ) + f x i+ ) n p H xi+ h p i x i p H ν h)p g s) ds, nd the corollry is proved xi+ xi+ g s) ds x i x i g s) ds H ν h)p p f t) g t) dt n xi+ x i g s) ds The previous corollry llows us to obtin dptive qudrture formule for different weighted integrls We point out only few exmples

370 8 SOME INEQUALITIES FOR RIEMANN-STIELTJES INTEGRAL 36 Exmple 5 Legendre) If g t), nd t, b, then we get the trpezoid formul for the mpping f :, b K of p H Hölder type: 837) where T f, I n ) is the usul trpezoid rule 838) T f, I n ) f t) dt T f, I n ) + R f, I n ), n nd the reminder stisfies the estimte f x i ) + f x i+ ) 839) R f, I n ) p H b ) ν h)p Exmple 6 Logrithm) If g t) ln ) t, t, b 0,, f is of p H Hölder type nd the integrl f t) ln ) t dt <, then we hve the generlized trpezoid formul: 840) f t) ln h i ) dt T L f, I n ) + R L f, I n ), t where T L f, I n ) is the following Logrithm-Trpezoid qudrture rule 84) T L f, I n ) n f x i ) + f x i+ ) xi ) x i ln x i+ ln e nd the reminder term R L f, I n ) stisfies the estimte 84) R L f, I n ) ) ) b p H ν h)p ln b ln e e xi+ Exmple 7 Jcobi) If g t) t, t, b 0, ), f is of p H Hölder type nd ft) t dt <, then we hve the generlized trpezoid formul 843) f t) t dt T J f, I n ) + R J f, I n ), where T J f, I n ) is the Jcobi-Trpezoid qudrture rule 844) T J f, I n ) n f x i ) + f x i+ ) xi+ ) x i nd the reminder term R J f, I n ) stisfies the estimte 845) R J f, I n ) p+ H ν ) h)p b Finlly, we hve Exmple 8 Chebychev) If g t) t, t, b, ), f is of p H Hölder type nd ft) t dt <, then we hve the generlized trpezoid formul 846) f t) t dt T c f, I n ) + R c f, I n ) e )

371 363 SS Drgomir where T c f, I n ) is the Chebychev-Trpezoid qudrture rule 847) T c f, I n ) : n f x i ) + f x i+ ) nd the reminder term R c f, I n ) stisfies the estimte rcsin x i+ ) rcsin x i ) 848) R c f, I n ) p H ν h)p rcsin b) rcsin ) 84 Another Trpezoid Like Inequlity The following theorem which complement in sense the previous result lso holds 35 Theorem 83 Let f :, b K be mpping of bounded vrition on, b nd u :, b K be p H Hölder type mpping, tht is, it stisfies the condition: 849) u x) u y) H x y p for ll x, y, b, where H > 0 nd p 0, re given Then we hve the inequlity: 850) Ψ f, u;, b) b p H b )p f) The constnt C on the right hnd side of 850) cnnot be replced by smller constnt Proof It is well known tht if g :, b K is continuous nd v :, b K is of bounded vrition, then the Riemnn-Stieltjes integrl g t) dv t) exists nd the following inequlity holds: b 85) g t) dv t) sup g t) v) t,b Using the integrtion by prts formul for Riemnn-Stieltjes integrls, we hve u ) + u b) u x) df x) u x) f ) + f b) u ) + u b) f x) u b) u ) b f x) du x) f x) du x) Consequently, if u nd f re s bove, then we hve the equlity: 85) f x) du x) f ) + f b) u b) u ) which is of importnce in itself too u x) u ) + u b) df x)

372 8 SOME INEQUALITIES FOR RIEMANN-STIELTJES INTEGRAL 364 Now, pplying the inequlity 85) for g t) u t) u)+ub) nd v t) f t), t, b, we obtin 853) Ψ f, u;, b) sup u ) + u b) b u t) f) t,b As u is of p H Hölder type, we hve u ) + u b) u t) u t) u ) + u t) u b) u t) u ) + u t) u b) H t )p + b t) p Now, consider the mpping ϕ t) : t ) p + b t) p, t, b, p 0, It is esy to see tht its mximum is relized for t +b nd mx t,b ϕ t) ϕ ) +b p b ) p Consequently, we hve sup t,b ) u ) + u b) p u t) b H Using 853), we deduce the desired inequlity 850) To prove the shrpness of the constnt, ssume tht 850) holds with constnt C > 0, ie, 854) Ψ f, u;, b) C b p H b )p f) Choose u x) x p, p 0,, x 0, which is of p Hölder type with the constnt H nd f : 0, R given by: { 0 if x 0, ), f x) if x which is of bounded vrition on 0, Substituting in 854), we obtin pt p f t) dt C p b ) f) However, 0 0 t p f t) dt 0 nd 0 f), nd then C p for ll p 0, Choosing p, we deduce C nd the theorem is completely proved 0 The following corollry is nturl consequence of the bove Theorem 83

373 365 SS Drgomir Corollry 84 Let f :, b K be s in Theorem 83 nd u be n L Lipschitzin mpping on, b, tht is, 855) u t) u s) L t s for ll t, s, b, where L > 0 is fixed Then we hve the inequlity 856) Ψ f, u;, b) L b b ) f) Remrk 8 If f :, b R is monotonic nd u is of p H Hölder type, then the inequlity 850) becomes: 857) Ψ f, u;, b) H b ) f b) f ) p In ddition, if u is L Lipschitzin, then the inequlity 856) cn be replced by 858) Ψ f, u;, b) L b ) f b) f ) Remrk 83 If f is Lipschitzin with constnt K > 0, then it is obvious tht f is of bounded vrition on, b nd b f) K b ) Consequently, the inequlity 850) becomes 859) Ψ f, u;, b) p HK b )p+, nd the inequlity 856) becomes 860) Ψ f, u;, b) LK b ) We now point out some results in estimting the integrl of product Corollry 85 Let f :, b R be mpping of bounded vrition on, b nd g be continuous on, b Put g : sup t,b g t) Then we hve the inequlity: f ) + f b) b 86) g s) ds f t) g t) dt g b b ) f) Remrk 84 Now, if in the bove corollry we ssume tht f is monotonic, then 86) becomes f ) + f b) b 86) g s) ds f t) g t) dt g f b) f ) b ), nd if in Corollry 85 we ssume tht f is K Lipschitzin, then the inequlity 86) becomes f ) + f b) b 863) g s) ds f t) g t) dt g K b ) The following corollry is lso nturl consequence of Theorem 83

374 8 SOME INEQUALITIES FOR RIEMANN-STIELTJES INTEGRAL 366 Corollry 86 Let f nd g be s in Corollry 85 Put ) b p g p : g s) p ds ; p > Then we hve the inequlity f ) + f b) 864) p p g p b ) p p g s) ds b f) f t) g t) dt Proof Consider the mpping u given by u t) : t g s) ds Then, by Hölder s integrl inequlity, we cn stte tht t u t) u s) g z) dz t s t q g z) p p dz s t s p p g p, for ll t, s, b, which shows tht the mpping u is of r H Hölder type with r : p p 0, ) nd H g p < The mpping u is lso differentible on, b) nd u t) g t), t, b) Therefore, by the theory of Riemnn-Stieltjes integrls, we hve f t) du t) s f t) g t) dt Using 850), we deduce the desired inequlity 864) We give now some exmples of weighted trpezoid inequlities for some of the most populr weights Exmple 9 Legendre) If gt), t, b then by 86) nd 864) we get the trpezoid inequlities 865) Ψ f;, b) b b ) f) nd 866) Ψ f;, b) b b ) f), p > /p We remrk tht the first inequlity is better thn the second one Exmple 0 Jcobi) If gt) t, t 0,, then obviously g +, so we cnnot pply the inequlity 86) If we ssume tht p, ) then we hve ) p /p ) /p g p dt t p 0

375 367 SS Drgomir nd pplying the inequlity 864) we deduce 867) f0) + f) ft)dt 0 t 4 p )/p p) /p for ll p, ) f), Exmple Chebychev) If gt), t, ), then obviously g t +, so we cnnot pply the inequlity 86) If we ssume tht p, ) then we hve ) p /p g p dt t t + ) p t) p p B /p, p ) /p /p dt Applying the inequlity 864) we deduce f ) + f) ft) 868) π dt t p B π, p ) /p f) for ll p, ) Approximtion of the Riemnn-Stieltjes Integrl Consider the prtition I n : x 0 < x < < x n < x n b of the intervl, b nd define h i : x i+ x i i 0,, n ), ν h) : mx {h i i {0,, n }} nd the sum 869) T n f, u, I n ) : n f x i ) + f x i+ ) u x i+ ) u x i ) The following pproximtion of the Riemnn-Stieltjes integrl lso holds 35 cf Theorem 89) Theorem 87 Let f :, b K be mpping of bounded vrition on, b nd u :, b K be p H Hölder type mpping Then we hve the qudrture formul 870) f t) du t) T n f, u, I n ) + R n f, u, I n ), where T n f, u, I n ) is the generlized trpezoid formul given by 869) nd the reminder R n f, u, I n ) stisfies the estimte 87) R n f, u, I n ) b p H ν h)p f)

376 8 SOME INEQUALITIES FOR RIEMANN-STIELTJES INTEGRAL 368 Proof We pply Theorem 83 on every subintervl x i, x i+ i 0,, n ) to obtin f x i ) + f x i+ ) xi+ 87) u x i+ ) u x i ) f t) du t) x i p H hp i x i+ x i f) Summing the inequlities 87) over i from 0 to n nd using the generlized tringle inequlity, we obtin R n f, u, I n ) n f x i ) + f x i+ ) n p H h p i x i+ x i b p H ν h)p f) nd the theorem is proved xi+ u x i+ ) u x i ) x i f) H ν h)p p n x i+ x i f) f t) du t) Remrk 85 Some prticulr results s in Corollries 80-8 nd Exmples 5-8 cn be stted s well, but we omit the detils 86 A Generlistion of the Trpezoid Inequlity The following theorem holds see 49) Theorem 88 Let u :, b R be of H r-hölder type, ie, we recll this 873) ux) uy) H x y r, for ny x, y, b nd some H > 0, where r 0, is given, nd f :, b R is of bounded vrition Then we hve the inequlity: 874) ft)dut) ub) ux))fb) + ux) u))f) H b ) + x + b r b b f) Hb ) r f) for ny x, b The constnt is shrp in the sense tht we cnnot put smller constnt insted 875) Proof Using the integrtion by prts formul, we my stte: ut) ux))dft) ub) ux)fb) u) ux)f) ft)dut)

377 369 SS Drgomir It is well known tht if m :, b R is continuous nd n :, b R is of bounded vrition, the Riemnn-Stieltjes integrl mt)dnt) exists, nd b mt)dnt) sup mt) n) Thus, Finlly, s ut) ux))dft) sup ut) ux) t,b b H mx{ b x r, x r } H b ) + we get the lst inequlity in 874) t,b b f) H sup { t x r } f) t,b b f) Hmxb x, x ) r x + b r b f) x + b b ) for ny x, b b f) To prove the shrpness of the constnt, we ssume tht 874) holds with the constnt c > 0, ie, 876) ft)dut) ub) ux))fb) + ux) u))f) H cb ) + x + b r b f) Choose ut) t which is of )-Hölder type nd f :, b R, ft) 0 if t {, b} nd ft) if t, b), which is of bounded vrition, in 876) We get: b cb ) + x + b, for ny x, b For x +b, we get: b cb ), ie c Remrk 86 If u is Lipschitz continuous function, ie ux) uy) L x y for ny x, y, b, nd some L > 0),

378 8 SOME INEQUALITIES FOR RIEMANN-STIELTJES INTEGRAL 370 the inequlity 874) becomes: 877) ft)dut) ub) ux))fb) + ux) u))f) L b ) + x + b b b f) Lb ) f) Corollry 89 If f is of bounded vrition on, b nd u is bsolutely continuous with u L, b then insted of L in 877) we cn put u ess sup u t) t,b Corollry 80 If g :, b R is Riemnn integrble on, b nd if we choose ut) t gs)ds, then x 878) ft)gt)dt fb) gs)ds f) gs)ds x g b ) + x + b b b f) g b ) f) Remrk 87 If in 878) we choose x +b, we get the best inequlity in the clss, ie, +b 879) ft)gt)dt fb) gs)ds f) gs)ds +b b g b ) f) 87 Approximting the Riemnn-Stieltjes Integrl Let I n : x 0 < x < < x n < x n b division of, b Denote h i : x i+ x i, nd νi n ) h i then construct the sums sup,n n n 880) Sf, u, I n, ξ) ux i+ ) uξ i )fx i+ ) + uξ i ) ux i )fx i ), where ξ i x i, x i+, i 0, n nd ξ ξ 0, ξ, ξ n ) We cn stte the following theorem concerning the pproximtion of Riemnn- Stieltjes integrl 49: Theorem 8 Let f, u be s in Theorem 88 nd I n, ξ s defined bove Then: 88) ft)dut) Sf, u, I n, ξ) + Rf, u, I n, ξ)

379 37 SS Drgomir when Sf, u, I n, ξ) is defined by 880) nd the reminder Rf, u, I n, ξ) stisfies the estimte: Rf, u, I n, ξ) H νi n) + sup ξ i x r i + x i+ b 88) f) H ν r I n ) b f),n Proof We pply 874) on x i, x i+ to get: xi+ ft)dut) ux i+ ) uξ i )fx i+ ) uξ i ) ux i )fx i ) x i H h i + ξ i x i + x i+ r x i+ x i+ f) H h r i f) Summing on i from 0 to n, nd using the generlised tringle inequlity we get: ft)dut) Sf, u, I n, ξ) n H h i + ξ i x i + x i+ r x i+ f) x i H sup,n h i + ξ x i + x i+ r b f) H νi n) + sup ξ i x r i + x i+ b f) Hν r I n ) nd the theorem is proved b f),,n Remrk 88 It is obvious tht if νi n ) 0 then 88) provides n pproximtion for the Riemnn-Stieltjes integrl ft)dut) Corollry 8 If we consider the sum then: 883) S M f, u, I n ) n ) xi + x i+ ux i+ ) u fx i+ ) + x i n u x i xi + x i+ ft)dut) S M f, u, I n ) + R M f, u, I n ) nd the reminder R M f, u, I n ) stisfies the estimte 884) R M f, u, I n ) b r Hνr I n ) f) ) ux i ) fx i )

380 8 SOME INEQUALITIES FOR RIEMANN-STIELTJES INTEGRAL 37 The following corollry in pproximting the integrl ft)gt)dt holds Corollry 83 If f, g re s in Corollry 80, then where n P f, g, I n, ξ) ft)gt)dt P f, g, I n, ξ) + R P f, g, I n, ξ) xi+ n fx i+ ) gs)ds + ξ i ξi fx i ) gs)ds x i nd the reminder R P f, g, I n, ξ) stisfies the estimte: R P f, g, I n, ξ) g νi n) + sup ξ i x i + x i+ b f) g νi n ) b f),n ) Remrk 89 If in the bove corollry we choose ξ i xi+xi+ i 0, n then we get the best formul in the clss, ie, nd n P M f, g, I n, ξ) fx i+ ) xi+ x i +x i+ R PM f, g, I n, ξ) g νi n) n x i +x i+ gs)ds + fx i ) gs)ds x i b f) 83 Inequlities of Ostrowski Type for the Riemnn-Stieltjes Integrl 83 Introduction In 938, A Ostrowski proved the following integrl inequlity see for exmple 70, p 468) Theorem 84 Let f :, b R be continuous on, b, differentible on, b), with its first derivtive f :, b) R bounded on, b), tht is, f : sup t,b) f t) < Then 885) f x) b f t) dt x +b 4 + b ) f b ), for ll x, b The constnt 4 is shrp in the sense tht it cnnot be replced by smller constnt For different proof thn the originl one provided by Ostrowski in 938 s well s pplictions for specil mens identric men, logrithmic men, p logrithmic men, etc) nd in Numericl Anlysis for qudrture formule of Riemnn type, see the recent pper 66 In 64, the following version of Ostrowski s inequlity for the -norm of the first derivtives hs been given

381 373 SS Drgomir Theorem 85 Let f :, b R be continuous on, b, differentible on, b), with its first derivtive f :, b) R integrble on, b), tht is, f : f t) dt < Then 886) f x) b f t) dt b + x +b f b, for ll x, b The constnt is shrp Note tht the shrpness of the constnt in the clss of differentible mppings whose derivtives re integrble on, b) hs been shown in the pper 7 In 64, the uthors pplied 886) for specil mens nd for qudrture formule of Riemnn type The following nturl extension of Theorem 85 hs been pointed out by SS Drgomir in 37 Theorem 86 Let f :, b R be mpping of bounded vrition on, b nd b f) its totl vrition on, b Then b 887) f x) b for ll x, b The constnt f t) dt is shrp + x +b b f), In 37, the uthor pplied 887) for qudrture formule of Riemnn type s well s for Euler s Bet mpping In wht follows we point out some generliztions of 887) for the Riemnn-Stieltjes integrl f t) du t) where f is of Hölder type nd u is of bounded vrition Applictions to the problem of pproximting the Riemnn-Stieltjes integrl in terms of Riemnn-Stieltjes sums re lso given 83 Some Integrl Inequlities The following theorem holds 4 Theorem 87 Let f :, b R be p H Hölder type mpping, tht is, it stisfies the condition 888) f x) f y) H x y p, for ll x, y, b ; where H > 0 nd p 0, re given, nd u :, b R is mpping of bounded vrition on, b Then we hve the inequlity 889) f x) u b) u )) f t) du t) H b ) + x + b p b u),

382 8 SOME INEQUALITIES FOR RIEMANN-STIELTJES INTEGRAL 374 for ll x, b, where b u) denotes the totl vrition of u on, b Furthermore, the constnt is the best possible, for ll p 0, Proof It is well known tht if g :, b R is continuous nd v :, b R is of bounded vrition, then the Riemnn-Stieltjes integrl g t) dv t) exists nd the following inequlity holds: b 890) g t) dv t) sup g t) v) t,b Using this property, we hve 89) f x) u b) u )) f t) du t) f x) f t)) du t) sup f x) f t) t,b As f is of p H Hölder type, we hve b u) sup f x) g t) sup H x t p t,b t,b H mx {x ) p, b x) p } H mx {x, b x} p H b ) + x + b p Using 89), we deduce 889) To prove the shrpness of the constnt for ny p 0,, ssume tht 889) holds with constnt C > 0, tht is, 89) f x) u b) u )) f t) du t) H C b ) + x + b p b u), for ll f, p H Hölder type mppings on, b nd u of bounded vrition on the sme intervl Choose f x) x p p 0, ), x 0, nd u : 0, 0, ) given by { 0 if x 0, ) u x) if x As f x) f y) x p y p x y p for ll x, y 0,, p 0,, it follows tht f is of p H Hölder type with the constnt

383 375 SS Drgomir By using the integrtion by prts formul for Riemnn-Stieltjes integrls, we hve: 0 f t) du t) f t) u t) 0 u t) df t) 0 nd u) 0 Consequently, by 89), we get x p C + x p, for ll x 0, For x 0, we get C + ) p, which implies tht C, nd the theorem is completely proved 0 The following corollries re nturl Corollry 88 Let u be s in Theorem 87 nd f :, b R n L Lipschitzin mpping on, b, tht is, L) f t) f s) L t s for ll t, s, b where L > 0 is fixed Then, for ll x, b, we hve the inequlity 893) where Θf, x;, b) L b ) + x + b b u) Θf, u; x,, b) f x) u b) u )) f t) du t) is the Ostrowski s functionl ssocited to f nd u s bove The constnt is the best possible Remrk 80 If u is monotonic on, b nd f is of p H Hölder type, then, by 889) we get 894) Θf, u; x,, b) H b ) + x + b p u b) u ), x, b, nd if we ssume tht f is L Lipschitzin, then 893) becomes 895) Θf, u; x,, b) L b ) + x + b u b) u ), x, b

384 8 SOME INEQUALITIES FOR RIEMANN-STIELTJES INTEGRAL 376 Remrk 8 If u is K Lipschitzin, then obviously u is of bounded vrition over, b nd b u) L b ) Consequently, if f is of p H Hölder type, then 896) nd it f is L Lipschitzin, then 897) Θf, u; x,, b) HK b ) + x + b p b ), x, b Θf, u; x,, b) LK b ) + x + b b ), x, b The following corollry concerning generliztion of the mid-point inequlity holds: Corollry 89 Let f nd u be s defined in Theorem 87 Then we hve the generlized mid-point formul 898) Υf, u; x,, b) H b p b )p u), where ) + b Υf, u; x,, b) f u b) u )) f t) du t) is the mid point functionl ssocited to f nd u s bove In prticulr, if f is L Lipschitzin, then 899) Υf, u; x,, b) L b b ) u) Remrk 8 Now, if in 898) nd 899) we ssume tht u is monotonic, then we get the midpoint inequlities 800) Υf, u; x,, b) H p b )p u b) u ) nd 80) Υf, u; x,, b) L b ) u b) u ) respectively In ddition, if in 898) nd 899) we ssume tht u is K Lipschitzin, then we obtin the inequlities 80) Υf, u; x,, b) HK b )p+ p nd 803) Υf, u; x,, b) LK b ) The following inequlities of rectngle type lso hold:

385 377 SS Drgomir Corollry 830 Let f nd u be s in Theorem 87 Then we hve the generlized left rectngle inequlity b 804) f ) u b) u )) f t) du t) H b )p u) nd the right rectngle inequlity 805) f b) u b) u )) b f t) du t) H b )p u) Remrk 83 If we dd the inequlities 804) nd 805), then, by using the tringle inequlity, we end up with the following generlized trpezoidl inequlity 806) f ) + f b) u b) u )) f t) du t) H b )p b u) In wht follows, we point out some results for the Riemnn integrl of product Corollry 83 Let f :, b R be p H Hölder type mpping nd g :, b R be continuous on, b Then we hve the inequlity f x) 807) g s) ds f t) g t) dt H b ) + x + b p g s) ds for ll x, b Proof Define the mpping u :, b R, u t) t g s) ds Then u is differentible on, b) nd u t) g t) Using the theory of the Riemnn-Stieltjes integrl, we hve nd b u) f t) du t) Now, from 889), we deduce 807) u t) dt f t) g t) dt g t) dt Remrk 84 The best inequlity we cn get from 807) is tht one for which x +b, obtining the midpoint inequlity ) + b b 808) f g s) ds f t) g t) dt H b )p g s) ds p We give some exmples of weighted Ostrowski inequlities for some of the most populr weights

386 8 SOME INEQUALITIES FOR RIEMANN-STIELTJES INTEGRAL 378 Exmple Legendre) If g t), nd t, b, then we get the following Ostrowski inequlity for Hölder type mppings f :, b R 809) b ) f x) f t) dt H b ) + x + b p b ) for ll x, b, nd, in prticulr, the mid-point inequlity ) + b b 80) b ) f f t) dt p H b )p+ Exmple 3 Logrithm) If g t) ln ) t, t 0,, f is of p Hölder type on 0, nd the integrl 0 f t) ln ) t dt is finite, then we hve ) 8) f x) f t) ln dt t H + x p for ll x 0, nd, in prticulr, ) 8) f 0 Exmple 4 Jcobi) If g t) 0 ft) t dt is finite, then we hve 83) f x) for ll x 0, nd, in prticulr, ) 84) f Finlly, we hve the following: 0 0 ) f t) ln dt t p H t, t 0,, f is s bove nd the integrl f t) dt t H + x 0 f t) dt t p H Exmple 5 Chebychev) If g t) t, t, ), f is of p Hölder type on, ) nd the integrl ft) t dt is finite, then 85) f x) f t) π dt H + x p t for ll x,, nd in prticulr, 86) f 0) π f t) t dt H 833 Approximtion of the Riemnn-Stieltjes Integrl Consider I n : x 0 < x < < x n < x n b to be division of the intervl, b, h i : x i+ x i i 0,, n ) nd ν h) : mx {h i i 0,, n } Define the generl Riemnn-Stieltjes sum n 87) S f, u, I n, ξ) : f ξ i ) u x i+ ) u x i )) p

387 379 SS Drgomir In wht follows, we point out some upper bounds for the error pproximtion of the Riemnn-Stieltjes integrl f t) du t) by its Riemnn-Stieltjes sum S f, u, I n, ξ) 4 Theorem 83 Let u :, b R be mpping of bounded vrition on, b nd f :, b R p H Hölder type mpping Then 88) f t) du t) S f, u, I n, ξ) + R f, u, I n, ξ), where S f, u, I n, ξ) is s given in 87) nd the reminder R f, u, I n, ξ) stisfies the bound R f, u, I n, ξ) H ν h) + mx ξ i x i + x i+ p b 89) u),n b H ν h) p u) Proof We pply Theorem 87 on the subintervls x i, x i+ i 0,, n ) to obtin xi+ f ξ 80) i) u x i+ ) u x i )) f t) du t) x i H h i + ξ i x i + x i+ p x i+ u), x i for ll i {0,, n } Summing over i from 0 to n nd using the generlized tringle inequlity, we deduce n xi+ R f, u, I n, ξ) f ξ i) u x i+ ) u x i )) f t) du t) However, sup,n n H h i + ξ i x i + x i+ p x i+ u) x i H sup,n h i + ξ i x i + x i+ p n x i+ u) x i h i + ξ i x i + x i+ p nd n x i+ b u) u), x i which completely proves the first inequlity in 89) For the second inequlity, we observe tht ξ i x i + x i+ h i, x i ν h) + sup ξ i x i + x i+ p

388 8 SOME INEQUALITIES FOR RIEMANN-STIELTJES INTEGRAL 380 for ll i {0,, n } The theorem is thus proved The following corollries re nturl Corollry 833 Let u be s in Theorem 83 nd f n L Lipschitzin mpping Then we hve the formul 88) nd the reminder R f, u, I n, ξ) stisfies the bound R f, u, I n, ξ) L ν h) + mx ξ i x i + x i+ b 8) u) Hν h) b u),n Remrk 85 If u is monotonic on, b, then the error estimte 89) becomes 8) nd 89) becomes 83) R f, u, I n, ξ) H ν h) + mx ξ i x i + x i+ p u b) u ),n H ν h) p u b) u ) R f, u, I n, ξ) L ν h) + mx ξ i x i + x i+ u b) u ),n Lν h) u b) u ) Using Remrk 8, we cn stte the following corollry Corollry 834 If u :, b R is Lipschitzin with the constnt K nd f :, b R is of p H Hölder type, then the formul 88) holds nd the reminder R f, u, I n, ξ) stisfies the bound n R f, u, I n, ξ) HK h i + ξ i x i + x i+ p 84) h i n HK h p+ i HK b ) ν h) p In prticulr, if we ssume tht f is L Lipschitzin, then R f, u, I n, ξ) n LK n h i + LK ξ i x i + x i+ 85) h i n LK h i LK b ) ν h) The best qudrture formul we cn get from Theorem 83 is tht one for which ξ i xi+xi+ for ll i {0,, n } Consequently, we cn stte the following corollry

389 38 SS Drgomir Corollry 835 Let f nd u be s in Theorem 83 Then 86) f t) du t) S M f, u, I n ) + R M f, u, I n ) where S M f, u, I n ) is the generlized midpoint formul, tht is; n ) xi + x i+ S M f, u, I n ) : f u x i+ ) u x i )) nd the reminder stisfies the estimte 87) R M f, u, I n ) H b p ν h)p u) In prticulr, if f is L Lipschitzin, then we hve the bound: 88) R M f, u, I n ) H b ν h) u) Remrk 86 If in 87) nd 88) we ssume tht u is monotonic, then we get the inequlities 89) R M f, u, I n ) H p ν h)p f b) f ) nd 830) R M f, u, I n ) H ν h) f b) f ) The cse where f is K Lipschitzin is embodied in the following corollry Corollry 836 Let u nd f be s in Corollry 834 Then we hve the qudrture formul 86) nd the reminder stisfies the estimte 83) R M f, u, I n ) HK p n h p+ i HK p ν h)p In prticulr, if f is L Lipschitzin, then we hve the estimte 83) R M f, u, I n ) n LK h i LK b ) ν h) 834 Another Inequlity of Ostrowski Type for the Riemnn-Stieltjes Integrl The following result holds 6: Theorem 837 Let f :, b R be mpping of bounded vrition on, b nd u :, b R be of r H Hölder type Then for ll x, b, we hve the

390 8 SOME INEQUALITIES FOR RIEMANN-STIELTJES INTEGRAL 38 inequlity 833) Θ f, u; x,, b) x b H x ) r f) + b x) r f) H x ) r + b x) r x b x f) + f) b f) ; x H x ) qr + b x) qr q if p >, p + q ; x p b ) p p f)) + f) x H b ) + x +b r b f) where Θ f, u; x,, b) u b) u )) f x) is the Ostrowski s functionl ssocited to f nd u s bove f t) du t) Proof As u is continuous nd f is of bounded vrition on, b, the following Riemnn-Stieltjes integrls exist nd, by the integrtion by prts formul, we cn stte tht 834) nd 835) x x u t) u )) df t) u x) u )) f x) u t) u b)) df t) u b) u x)) f x) If we dd the bove two identities, we obtin 836) Θ f, u; x,, b) x u t) u )) df t) + for ll x, b, which is of importnce in itself Now, using the properties of modulus, we hve: x x x u t) u b)) df t) f t) du t) f t) du t) 837) Θ f, u; x,, b) x u t) u )) df t) + sup u t) u ) t,x x x u t) u b)) df t) f) + sup u t) u b) t x,b b f), x

391 383 SS Drgomir nd for the lst inequlity we hve used the well-known property of Riemnn- Stieltjes integrls, ie, d d 838) p t) dv t) sup p t) v), c t c,d provided tht p is continuous on c, d nd v is of bounded vrition on c, d Now, s u is of r H Hölder type on, b, we cn stte tht nd then nd u t) u ) H t ) r, u t) u b) H b t) r sup u t) u ) H x ) r t,x sup u t) u b) H b x) r t x,b Now, using 837), we cn stte tht x b Θ f, u; x,, b) H x ) r f) + b x) r f), for ll x, b, nd the first inequlity in 833) is proved Further on, define the mpping M :, b R, given by M x) x ) r x c f) + b x) r x b f) It is obvious tht { x } b M x) mx f), f) x ) r + b x) r x x b x b f) + f) + f) f) x ) r + b x) r x x b f) + x b f) f) x ) r + b x) r, nd the first prt of the second inequlity is thus proved Using the elementry inequlity of Hölder type, we obtin 0 αβ + γδ α p + γ p ) p β q + δ q ) q, α, β, γ, δ 0, p >, x x p + q, x p b p M x) x ) qr + b x) qr q f) p + f), x

392 8 SOME INEQUALITIES FOR RIEMANN-STIELTJES INTEGRAL 384 nd the second prt of the second inequlity is proved Finlly, we observe tht x M x) mx {x ) r, b x) r } f) + b mx {x, b x} r f) b ) + x + b r b f), b f) x nd the lst prt of the second inequlity is proved The following corollries re nturl consequences of 833) Corollry 838 Let f be s in Theorem 837 nd u :, b R be n L Lipschitzin mpping on, b Then, for ll x, b, we hve the inequlity 839) Θ f, u; x,, b) x L x ) f) + b x) b f) x L b f) + x f) b f) b ) ; L x ) q + b x) q q if p >, p + q ; x x p b ) p p f)) + f) x L b ) + x +b b f) Remrk 87 If f :, b R is monotonic on, b nd u is of r H Hölder type, then f is of bounded vrition on, b, nd by 833) we obtin 840) Θ f, u; x,, b) H x ) r f x) f ) + b x) r f b) f x) H f f b) f ) + x) f)+fb) b x) r + x ) r ; H f x) f ) p + f b) f x) p p b x) qr + x ) qr q if p >, p + q ; H f b) f ) b ) + x +b r,

393 385 SS Drgomir for ll x, b In prticulr, if u is L Lipschitzin on, b, then from 839) we get 84) Θ f, u; x,, b) L x ) f x) f ) + b x) f b) f x) L f f b) f ) + x) f)+fb) b ) ; L f x) f ) p + f b) f x) p p x ) q + b x) q q, p >, p + q ; L f b) f ) b ) + x +b, for ll x, b Remrk 88 If f :, b R is Lipschitzin with constnt K > 0, then, obviously f is of bounded vrition on, b nd b f) K b ) Consequently, from the first inequlity in 833), we deduce 84) Θ f, u; x,, b) HK x ) r+ + b x) r+ for ll x, b If we ssume tht f is L Lipschitzin, then from 84) we get 843) Θ f, u; x,, b) LK b ) + x + b ), for ll x, b The following corollry concerning generliztion of the mid-point inequlity holds Corollry 839 Let f nd u be s in Theorem 837 Then we hve the inequlity 844) Υ f, u; x,, b) r H b f) b ) r where ) + b b Υ f, u; x,, b) f u b) u )) f t) du t) is the mid point functionl ssocited to f nd u s bove In prticulr, if u is L Lipschitzin, then b 845) Υ f, u; x,, b) L f) b ) Remrk 89 Now, if in 843) nd 844) we ssume tht f is monotonic, then we get the inequlities 846) Υ f, u; x,, b) r H f b) f ) b ) r nd 847) Υ f, u; x,, b) L f b) f ) b )

394 8 SOME INEQUALITIES FOR RIEMANN-STIELTJES INTEGRAL 386 Also, if we ssume tht in 843) nd 844) f is K Lipschitzin, then we get the inequlities 848) Υ f, u; x,, b) r HK b ) r+ nd 849) Υ f, u; x,, b) LK b ) Another interesting corollry is the following one Corollry 840 Let f nd u be s in Theorem 837 If x 0, b is point stisfying the property tht x 0 b E) f) f) x 0 then we get the inequlity 850) f x 0) u b) u )) f t) du t) b H f) x 0 ) r + b x 0 ) r In prticulr, if u is L Lipschitzin on, b, then we hve 85) f x 0) u b) u )) f t) du t) b L b ) f) Remrk 80 If in 849) nd 850), we ssume tht f is monotonic nd x 0 is point such tht f x 0 ) f)+fb), then we hve the inequlity: f x 85) 0) u b) u )) f t) du t) H f b) f ) x 0 ) r + b x 0 ) r nd 853) f x 0) u b) u )) f t) du t) L b ) f b) f ) The following inequlities of rectngle type lso hold Corollry 84 Assume tht f nd u re s in Theorem 837 Then we hve the generlized left rectngle inequlity: b 854) f ) u b) u )) f t) du t) H b )r f) We lso hve the right rectngle inequlity b 855) f b) u b) u )) f t) du t) H b )r f)

395 387 SS Drgomir Remrk 8 If we dd the inequlities 854) nd 855), nd use the tringle inequlity, we end up with the following generlized trpezoid inequlity 856) f ) + f b) u b) u )) b f t) du t) H b )r f) Now, we point out some results for the Riemnn integrl of product Corollry 84 Let f :, b R be mpping of bounded vrition on, b nd g be continuous on, b Put g : sup t,b g t) Then we hve the inequlity f x) 857) g s) ds f t) g t) dt x b g x ) f) + b x) f) g b ) b f) + x x f) b f) ; x g x ) q + b x) q q if p >, p + q ; x g b ) + x +b b f), p b ) p p f)) + f) x for ll x, b Similr results for f monotonic of f Lipschitzin with constnt K > 0 pply, but we omit the detils Finlly, we hve Corollry 843 Let f nd g be s in Corollry 84 Put g p : g s) p ds ) p, p > Then we hve the inequlity 858) f x) g p g s) ds x ) p p x f t) g t) dt f) + b x) p p b f) x

396 8 SOME INEQUALITIES FOR RIEMANN-STIELTJES INTEGRAL 388 g p x ) p p g p x ) αp ) p where α >, α + β ; + b x) p p + b x) αp ) α x p g p b ) + x +b p p b x f) + f) b f) ; b f), x β b ) β β f)) + f), x for ll x, b Similr results for f monotonic nd f Lipschitzin with constnt K > 0 pply, but we omit the detils Corollry 844 Let f be of bounded vrition on, b nd g L, b Put g : g t) dt Then we hve the inequlity 859) for ll x, b f x) sup t,x x t g g s) ds g s) ds f t) g t) dt x g s) ds f) + sup x f) + x t x,b t g s) ds b f) + x b f) f), x b g s) ds f) b f) x x Proof Put in inequlity 837) u t) t g s) ds, to obtin f x) sup t,x sup t t t,x t sup t,x g s) ds f t) g t) dt x g s) ds f) + sup t x,b x g s) ds f) + sup g s) ds x t x,b f) + sup t x,b t t t g s) ds g s) ds b g s) ds f) x g s) ds b f) x b f) x

397 389 SS Drgomir x g s) ds x f) + x g s) ds { x } b x mx f), f) g s) ds + g nd the corollry is proved x b f) + x b f) f), x b f) x x g s) ds 835 Approximtion of the Riemnn-Stieltjes Integrl Consider I n : x 0 < x < < x n < x n b division of the intervl, b, h i : x i+ x i i 0,, n ) nd ν h) : mx {h i i 0,, n } Define the generl Riemnn-Stieltjes sum s n 860) S f, u, I n, ξ) : f ξ i ) u x i+ ) u x i )) In wht follows, we point out some upper bounds for the error of the pproximtion of the Riemnn-Stieltjes integrl f t) du t) by its Riemnn-Stieltjes sum S f, u, I n, ξ) Every inequlity in Theorem 837 cn be used in pointing out n upper bound for tht error However, we feel tht the lst inequlity cn give much simpler nd nicer one Therefore, we will focus our ttention to tht one lone nd its consequences 6 Theorem 845 Let f :, b R be mpping of bounded vrition on, b nd u :, b R be of r H Hölder type on, b Then 86) f t) du t) S f, u, I n, ξ) + R f, u, I n, ξ) where S f, u, I n, ξ) is s given in 860) nd the reminder R f, u, I n, ξ) stisfies the bound 86) R f, u, I n, ξ) H ν h) + mx ξ i x i + x i+ r b f),n Proof We pply Theorem 837 on the subintervls x i, x i+ i 0,, n ) to get xi+ f ξ 863) i) u x i+ ) u x i )) f t) du t) x i H h i + ξ i x i + x i+ r x i+ f) x i

398 8 SOME INEQUALITIES FOR RIEMANN-STIELTJES INTEGRAL 390 for ll i {0,, n } Summing over i from 0 to n nd using the generlized tringle inequlity, we deduce However, nd R f, u, I n, ξ) sup,n h i + n f ξ i) u x i+ ) u x i )) xi+ x i f t) du t) n H h i + ξ i x i + x i+ r x i+ f) x i H sup,n h i + ξ i x i + x i+ r n x i+ f) x i ξ i x i + x i+ r n x i+ x i f) ν h) + sup b f),,n which completely proves the first inequlity in 86) For the second inequlity, we observe tht ξ i x i + x i+ h i, for ll i {0,, n } The theorem is thus proved ξ i x i + x i+ r The following corollries re nturl consequence of Theorem 845 Corollry 846 Let f be s in Theorem 845 nd u be n L Lipschitzin mpping Then we hve the formul 86), where the reminder R f, u, I n, ξ) stisfies the bound 864) R n f, u, I n, ξ) L ν h) + mx ξ i x i + x i+ b f) Hν h) b f),n Remrk 8 If f is monotonic on, b, then the error estimte 86) becomes 865) R n f, u, I n, ξ) H ν h) + mx ξ i x i + x i+ f b) f ),n H ν h) r f b) f )

399 39 SS Drgomir nd 864) becomes 866) R n f, u, I n, ξ) L ν h) + mx ξ i x i + x i+ f b) f ),n Lν h) f b) f ) Using Remrk 88, we cn stte the following corollry Corollry 847 Let u be s in Theorem 845 nd f :, b R be K Lipschitzin mpping on, b Then the pproximtion formul 86) holds nd the reminder R f, u, I n, ξ) stisfies the bound 867) n R n f, u, I n, ξ) HK ξ i x i ) r+ + x i+ ξ i ) r+ n HK h r+ i HK b ) ν h) r In prticulr, if u is L Lipschitzin, we get n n R n f, u, I n, ξ) LK h i + ξ i x ) i + x i+ 868) n LK h i LK b ) ν h) We hve the mention tht the best inequlity we cn get in Theorem 845 is tht one for which ξ i xi+xi+ for ll i {0,, n } Consequently, we cn stte the following corollry: Corollry 848 Let f nd u be s in Theorem 845 Then 869) f t) du t) S M f, u, I n ) + R M f, u, I n ) where S M f, u, I n ) is the generlized mid-point formul Tht is, n ) xi + x i+ S M f, u, I n ) : f u x i+ ) u x i )) nd the reminder R M f, u, I n ) stisfies the estimte 870) R M f, u, I n ) H b r ν h)r f) In prticulr, if u is L Lipschitzin, then we hve the bound: 87) R M f, u, I n ) H b ν h) f)

400 8 SOME INEQUALITIES FOR RIEMANN-STIELTJES INTEGRAL 39 Remrk 83 If in 870) nd 87) we ssume tht f is monotonic, then we get the inequlities 87) R M f, u, I n ) H r ν h)r f b) f ) nd 873) R M f, u, I n ) H ν h) f b) f ) The cse where f is K Lipschitzin is embodied in the following corollry Corollry 849 Let u nd f be s in Corollry 847 Then we hve the qudrture formul 869) where the reminder stisfies the estimte 874) R M f, u, I n ) HK r n h r+ i HK r b ) ν h) r In prticulr, if u is Lipschitzin, then we hve the estimte 875) R M f, u, I n ) n LK h i LK b ) ν h) 84 Some Inequlities of Grüss Type for Riemnn-Stieltjes Integrl 84 Introduction In 935, G Grüss 68 proved n inequlity which estblishes connection between the integrl of the product of two functions nd the product of the integrls Nmely, he hs shown tht: b f x) g x) dx b f x) dx b g x) dx Φ φ) Γ γ), 4 provided f nd g re two integrble functions on, b nd stisfy the condition: for ll x, b The constnt 4 φ f x) Φ nd γ g x) Γ is the best possible one nd is chieved for f x) g x) sgn x + b ) In the recent pper 56, SS Drgomir nd I Fedotov proved the following results of Grüss type for Riemnn-Stieltjes integrl: Theorem 850 Let f, u :, b R be such tht u is L-Lipschitzin on, b, ie, 876) ux) uy) L x y for ll x, y, b, f is Riemnn integrble on, b nd there exists the rel numbers m, M so tht 877) m fx) M,

401 393 SS Drgomir for ll x, b Then we hve the inequlity ub) u) 878) fx)dux) b ft)dt LM m)b ), nd the constnt is shrp, in the sense it cn not be replced by smller one For other results relted to Grüss inequlity, generliztions in inner product spces, for positive liner functionls, discrete versions, determinntl versions, etc, see Chpter X of the book 69 nd the ppers -67, where further references re given In this section we point out n inequlity of Grüss type for Lipschitzin mppings nd functions of bounded vrition s well s its pplictions in numericl integrtion for the Riemnn-Stieltjes integrl 84 Integrl Inequlities The following result of Grüss type holds 55 Theorem 85 Let f, u :, b R be such tht u is L-Lipschitzin on, b, nd f is function of bounded vrition on, b Then the following inequlity holds f b) f ) b 879) u x) df x) u t) dt b L b b ) f) The constnt is shrp, in the sense tht it cnnot be replced by smller one Proof As f is function of bounded vrition on, b nd u is continuous on, b, we hve f b) f ) b u x) df x) u t) dt b b u x) ) b u t) dt df x) b sup u x) b u t) dt f) b x,b 880) b sup x,b b u x) u t) dt f) Using the fct tht u is L Lipschitzin on, b, we cn stte, for ny x, b, tht: u x) u t) dt u x) u t) dt L x t dt L x ) + x b)

402 8 SOME INEQUALITIES FOR RIEMANN-STIELTJES INTEGRAL ) L 4 + x +b b ) nd by 880)-88) we get: 88) sup u x) b x,b ) b ) L b ), u t) dt L b ) whence we obtin 879) To prove the shrpness of the inequlity 879), let us choose u x) x + b if x nd f x) 0 if < x < b if x b Then u is Lipschitzin with L, nd f is of bounded vrition Also we hve u x) df x) x + b x + b f b) f ) b ) df x) ) f x) b u t) dt f x) dx b On the other hnd, the right hnd side of 879) is equl to b, nd hence the shrpness of the constnt is proved The following corollries hold: Corollry 85 Let f :, b R be s bove nd u :, b R be differentible mpping with bounded derivtive on, b Then we hve the inequlity: 883) u x) df x) f b) f ) b u t) dt u b ) b f) The inequlity 883) is shrp in the sense tht the constnt cnnot be replced by smller one Corollry 853 Let u be s bove nd f :, b R be differentible mpping whose derivtive is integrble, ie, f Then we hve the inequlity: 884) u x) f x) dx u f b ) f t) dt < f b) f ) b u t) dt

403 395 SS Drgomir Remrk 84 If we ssume tht g :, b R is continuous on, b nd if we set f x) x g x) dx, then from 884) we get the following Grüss type inequlity for the Riemnn integrl: b u x) g x) dx b 885) u t) dt g t) dt b b b u g b ) Corollry 854 Assume tht u :, b R is differentible on, b), u ) u b), nd u :, b R is bounded on, b) Then we hve the trpezoid inequlity: 886) u ) + u b) b ) u t) dt u u u b) u ) b ) Proof If we choose in Corollry 853, f x) u x), we get u x) u u b) u ) b 887) x) dx u t) dt b u u Now 886) follows from 887) b ) 843 A Numericl Qudrture Formul for the Riemnn-Stieltjes Integrl In wht follows, we shll pply Theorem 85 to pproximte the Riemnn- Stieltjes integrl u x) df x) in terms of the Riemnn integrl u t) dt see lso 55) Theorem 855 Let f, u :, b R be s in Theorem 85 nd I n { x 0 < x < < x n < x n b} prtition of, b Denote h i x i+ x i, i 0,,, n Then we hve: 888) where 889) A n u, f, I n ) u x) df x) A n u, f, I n ) + R n u, f, I n ) n f x i+ ) f x i ) xi+ u t) dt h i x i nd the reminder term R n u, f, I n ) stisfies the estimtion 890) R n u, f, I n ) L b ν h) f) where ν h) mx {h i},,n

404 8 SOME INEQUALITIES FOR RIEMANN-STIELTJES INTEGRAL 396 get Proof Applying Theorem 85 on the intervl x i, x i+, i 0,,, n we xi+ x i u x) df x) f x i+) f x i ) h i xi+ x i u t) dt L x h i+ i f) x i Summing over i from 0 to n nd using the tringle inequlity we obtin u x) df x) A n u, f, I n ) L n x i+ h i f) L n x ν h) i+ f) nd the corollry is proved x i L b ν h) f) Remrk 85 Consider the equidistnt prtition I n given by nd define A n u, f, I n ) n f n b Then we hve I n : x i + i b, i 0,, n; n + i + ) b ) f + i b ) +i+) b n u t) dt n n +i b n u x) df x) A n u, f) + R n u, f), where the reminder R n u, f) stisfies the estimtion R n u, f) L b ) n b f) Thus, if we wnt to pproximte the integrl u x) df x) by the sum A n u, f, I n ) with n error of mgnitude less thn ε we need t lest L b ) b n 0 f) + N ε points Corollry 856 Assume tht u nd f re s in Corollry 85 If I n is s bove, then 888) holds nd the reminder term R n u, f, I n ) stisfies the estimtion 89) R n u, f, I n ) u ν h) b f) Corollry 857 Assume tht u nd f re s in Corollry 853 Then 888) holds nd the reminder term R n u, f, I n ) stisfies the estimtion 89) R n u, f, I n ) u f ν h) x i

405 397 SS Drgomir 844 Qudrture Methods for the Riemnn-Stieltjes Integrl of Continuous Mppings Let I n { x 0 < x < < x n < x n b} be prtition of, b nd denote h i x i+ x i, i 0,, n We strt to the following lemm which is of interest in itself 55 Lemm 858 Let f be function from C x i, x i+, ie, f is continuous on x i, x i+, nd let u be function of bounded vrition on the sme intervl The following inequlity holds: 893) xi+ x i f x) du x) u x i+ ) u x i ) f i ω f, h i where f i xi+ h x i f x) dx, h i x i+ x i, nd ω f, h i sup f x) f t), i x t δ is the modulus of continuity x i+ x i u) Proof Since f Cx i, x i+ nd u is function of bounded vrition, by the well known property of such couple of functions we hve xi+ x f x) du x) f i+ Cx i,x i+ u) x i x i Therefore, xi+ x i xi+ x i f x) fi du x) f x) du x) u x i+ ) u x i ) f i f x) fi Cxi,x i+ x i+ x i sup x i+ f x) fi x x i,x i+ x i u) u) x i+ sup h i f x) h i fi u) h i x x i,x i+ x i xi+ xi+ sup h i f x) dt f t) dt x x i,x i+ x i x i xi+ x i+ sup h i f x) f t) dt u) x x i,x i+ x i x i+ sup f x) f t) u) x x i,x i+,t x i,x i+ x i+ ω f, h i u), x i nd the lemm is proved x i x i x i+ x i u)

406 8 SOME INEQUALITIES FOR RIEMANN-STIELTJES INTEGRAL 398 Let f C, b The dul to C, b is the spce of functions of bounded vrition, the generl form of the functionl on C, b is I f) f x) du x) where u belongs to the spce of functions of bounded vrition Tht is why in the theory of qudrture methods for continuous functions the cse of the integrls of such type is the most interesting For the integrl with continuous integrnd f x) w x) dx w x) > 0) we introduce the error functionl for the qudrture rule with the weights s nk nd nodes x k ) I f) by the formul f t) w t) dt k0 E n f) I f) I n f) n f x k ) s nk I n f) Here is the more generl result for the composite qudrture rules for the functions from C, b, 55 Theorem 859 Let nd b be finite rel numbers nd let f, u :, b R be such tht f C, b nd u is function of bounded vrition on, b If x 0 < x < < x n < x n b is division of, b such tht h i < δ, for ll i 0,,, n where h i x i+ x i, then the following estimtion for the Error functionl of the Riemnn-Stieltjes qudrture rule is true 894) E comp n f) ω f, δ where E comp n f) n f x) du x) b u), u x i+ ) u x i ) xi+ f x) dx, h i x i nd ω f, δ is the modulus of continuity of f with respect to δ Proof For given division of, b s bove, we hve n f x) du x) xi+ x i f x) du x) Then by Lemm 858, we cn write successively:

407 399 SS Drgomir En comp n n f) n f x) du x) xi+ x i xi+ n f x) du x) u x i+ ) u x i ) xi+ h i x i f x) du x) u x i+) u x i ) x i h i n x i+ ω f, h i u) ω f, δ x i ω f, δ b u), nd the theorem is proved u x i+ ) u x i ) xi+ h i x i n x i+ x i u) xi+ x i f x) dx f x) dx f x) dx

408

409 Bibliogrphy NS BARNETT nd SS DRAGOMIR, An inequlity of Ostrowski s type for cumultive distribution functions, Kyungpook Mth J, 39 ) 999), NS BARNETT nd SS DRAGOMIR, An Ostrowski type inequlity for double integrls nd pplictions for cubture formule, Soochow J of Mth, ccepted 3 NS BARNETT, P CERONE, SS DRAGOMIR, J ROUMELIOTIS, nd A SOFO, A survey on Ostrowski type inequlities for twice differentible mppings nd ppliction, submitted 4 N S BARNETT, SS DRAGOMIR nd C E M PEARCE, A qusi-trpezoid inequlity for double integrls, ANZIAM Journl, ccepted 5 R G BARTLE, The Elements of Rel Anlysis, Second Edition, John Wiley & Sons Inc, P CERONE nd SS DRAGOMIR, Lobto Type Qudrture Rules for Functions with Bounded Derivtives, Mth Ineq & Appl, 3 ) 000), P CERONE nd SS DRAGOMIR, Midpoint-type rules from n inequlities point of view, Hndbook of Anlytic-Computtionl Methods in Applied Mthemtics, Editor: G Anstssiou, CRC Press, New York 000), P CERONE nd SS DRAGOMIR, On weighted generliztion of iyengr type inequlities involving bounded first derivtive, Mth Ineq & Appl, 3 ) 000), P CERONE nd SS DRAGOMIR, Three point qudrture rules involving, t most, first derivtive, submitted 0 P CERONE nd SS DRAGOMIR, Trpezoidl-type rules from n inequlities point of view, Hndbook of Anlytic-Computtionl Methods in Applied Mthemtics, Editor: G Anstssiou, CRC Press, New York 000), P CERONE, SS DRAGOMIR nd J ROUMELIOTIS, An inequlity of Ostrowski-Grüss type for twice differentible mppings nd pplictions, Kyungpoook Mth J, 39 ) 999), P CERONE, SS DRAGOMIR nd J ROUMELIOTIS, An inequlity of Ostrowski type for mppings whose second derivtives re bounded nd pplictions, Est Asin J of Mth, 5 ) 999), -9 3 P CERONE, SS DRAGOMIR nd J ROUMELIOTIS, An inequlity of Ostrowski type for mppings whose second derivtives belong to L nd pplictions, Honm Mth J, ) 999), P CERONE, SS DRAGOMIR nd J ROUMELIOTIS, An Ostrowski type inequlity for mppings whose second derivtives belong to L p nd pplictions, submitted 5 P CERONE, SS DRAGOMIR nd J ROUMELIOTIS, Some Ostrowski type inequlities for n-time differentible mppings nd pplictions, Demonstrtio Mth, 3 ) 999), SS DRAGOMIR, A generliztion of Grüss inequlity in inner product spces nd pplictions, J Mth Anl Appl, ), SS DRAGOMIR, A generliztion of Ostrowski integrl inequlity for mppings of bounded vrition nd pplictions in numericl integrtion, submitted 8 SS DRAGOMIR, A generliztion of Ostrowski integrl inequlity for mppings whose derivtives belong to L nd pplictions in numericl integrtion, SUT Mth J, ccepted 9 SS DRAGOMIR, A generliztion of Ostrowski integrl inequlity for mppings whose derivtives belong to L, b nd pplictions in numericl nlysis, J of Computtionl Anlysis nd Appl, in press 40

410 BIBLIOGRAPHY 40 0 SS DRAGOMIR, A generliztion of Ostrowski integrl inequlity for mppings whose Derivtives Belong to L p, b nd pplictions in numericl nlysis, J Mth Anl Appl, ccepted SS DRAGOMIR, A Grüss type integrl inequlity for mppings of r-hölder s type nd pplictions for trpezoid formul, Tmkng J of Mth, 3 ) 000), SS DRAGOMIR, Grüss inequlity in inner product spces, The Austrlin Mth Soc Gzette, 6) 999), SS DRAGOMIR, On the Ostrowski s inequlity for mppings of bounded vrition, submitted 4 SS DRAGOMIR, On the Ostrowski inequlity for the Riemnn-Stieltjes integrl I), in preprtion 5 SS DRAGOMIR, On the Ostrowski inequlity for the Riemnn-Stieltjes integrl II), in preprtion 6 SS DRAGOMIR, On the Ostrowski s inequlity for Riemnn-Stieltjes integrl nd pplictions, Koren J Appl Mth, 7 000), SS DRAGOMIR, On Simpson s qudrture formul for differentible mppings whose derivtives belong to L p -spces nd pplictions, JKSIAM, ) 998), SS DRAGOMIR, On Simpson s qudrture formul nd pplictions, submitted 9 SS DRAGOMIR, On Simpson s qudrture formul for Lipschitzin mppings nd pplictions, Tmkng J of Mth, 30) 999), SS DRAGOMIR, On Simpson s qudrture formul for mppings with bounded vrition nd ppliction, Soochow J of Mth, 5) 999), SS DRAGOMIR, On the Ostrowski integrl inequlity for Lipschitzin mppings nd pplictions, Computer nd Mth with Appl, ), SS DRAGOMIR, On the Ostrowski s integrl inequlity for mppings with bounded vrition nd pplictions, Bull Austrl Mth Soc, ), SS DRAGOMIR, On the trpezoid formul for Lipschitzin mppings nd pplictions, Tmkng J of Mth, 30) 999), SS DRAGOMIR, On the trpezoid inequlity for bsolutely continuous mppings, submitted 35 SS DRAGOMIR, On the trpezoid inequlity for the Riemnn-Stieltjes integrl nd pplictions, submitted 36 SS DRAGOMIR, On the trpezoid qudrture formul for mppings of bounded vrition nd pplictions, Extrct Mth, ccepted 37 SS DRAGOMIR, Ostrowski s inequlity for mppings of bounded vrition nd pplictions, RGMIA Res Rep Coll, ) 999), 03-0, Mth Ineq & Appl, ccepted) 38 SS DRAGOMIR, Ostrowski s inequlity for monotonic mpping nd pplictions, J KSIAM, 3 ) 999), SS DRAGOMIR, Some discrete inequlities of Grüss type nd pplictions in guessing theory, submitted 40 SS DRAGOMIR, Some integrl inequlities of Grüss type, Indin J of Pure nd Appl Mth, 3 4) 000), SS DRAGOMIR, Some qudrture formule for bsolutely continuous mppings vi Ostrowski type inequlities, submitted 4 SS DRAGOMIR, RP AGARWAL nd NS BARNETT, Inequlities for bet nd gmm functions vi some clssicl nd new integrl inequlities, Journl of Inequlities nd Applictions, 5 000), SS DRAGOMIR, RP AGARWAL nd P CERONE, On Simpson s inequlity nd pplictions, Journl of Inequlities nd Applictions, ccepted 44 SS DRAGOMIR nd NS BARNETT, An Ostrowski type inequlity for mppings whose second derivtives re bounded nd pplictions, Kyungpoook Mth J, ccepted 45 SS DRAGOMIR, NS BARNETT nd P CERONE, An n-dimensionl version of Ostrowski s inequlity for mppings of the Hölder s type, Kyungpoook Mth J, 40) 000), SS DRAGOMIR, NS BARNETT nd PCERONE, An Ostrowski type inequlity for double integrls in terms of L p norms nd pplictions in numericl integrtion, Anl Num Theor Approx, ccepted

411 403 SS Drgomir 47 SS DRAGOMIR, NS BARNETT nd S WANG, An Ostrowski type inequlity for rndom vrible whose probbility density function belongs to L p, b, p >, Mthemticl Inequlities nd Appl, 4) 999), SS DRAGOMIR nd GL BOOTH, On Grüss-Lupş type inequlity nd its pplictions for the estimtion of p-moments of guessing mppings, Mthemticl Communictions, ccepted 49 SS DRAGOMIR, C BUSE, VM BOLDEA nd L BRAESCU, A generlistion of the trpezoid rule for the Riemnn-Stieltjes integrl nd pplictions, RGMIA Res Rep Coll, 34) 000), Article 50 SS DRAGOMIR, PCERONE, NS BARNETT nd J ROUMELIOTIS, An inequlity of the Ostrowski type for double integrls nd pplictions for cubture formule, Tmsui Oxford J Mth Sci, 6 ) 000), -6 5 SS DRAGOMIR, P CERONE nd C E M PEARCE, Generliztions of the trpezoid inequlity for mppings of bounded vrition nd pplictions, Turkish J of Mth, in press) 5 SS DRAGOMIR, PCERONE nd J ROUMELIOTIS, A new generliztion of Ostrowski s integrl inequlity for mppings whose derivtives re bounded nd pplictions in numericl integrtion, Appl Mth Lett, 3 000), SS DRAGOMIR, PCERONE, J ROUMELIOTIS nd S WANG, A weighted version of Ostrowski inequlity for mppings of Hölder type nd pplictions in numericl nlysis, Bull Mth Soc Mth Romni, 4 90)4) 999), SS DRAGOMIR, P CERONE nd A SOFO, Some remrks on the trpezoid rule in numericl integrtion, Indin J of Pure nd Appl Mth, 35) 000), SS DRAGOMIR nd I FEDOTOV, A Grüss type inequlity for mppings of bounded vrition nd pplictions to numericl nlysis, submitted 56 SS DRAGOMIR, nd I FEDOTOV, An inequlity of Grüss type for Riemnn-Stieltjes integrl nd pplictions for specil mens, Tmkng J of Mth, 94) 998), SS DRAGOMIR, A KUCERA nd J ROUMELIOTIS, A trpezoid formul for Riemnn- Stieltjes integrl, in preprtion 58 SS DRAGOMIR nd A McANDREW, On trpezoid inequlity vi Grüss type result nd pplictions, Tmkng J of Mth, in press 59 SS DRAGOMIR nd T C PEACHEY, New estimtion of the reminder in the trpezoidl formul with pplictions, Studi Mth Bbes-Bolyi Univ, ccepted 60 SS DRAGOMIR nd CEM PEARCE, Selected Topics on Hermite-Hdmrd Inequlities nd Applictions, RGMIA Monogrphs, Victori University, 000 ONLINE: 6 SS DRAGOMIR, J E PEČARIĆ nd S WANG, The unified tretment of trpezoid, Simpson nd Ostrowski type inequlity for monotonic mppings nd pplictions, Computer nd Mth with Appl, 3 000), SS DRAGOMIR nd A SOFO, An estimtion for ln k, Austrlin Mth Gzette, 65) 999), SS DRAGOMIR nd A SOFO, An integrl inequlity for twice differentible mppings nd pplictions, Tmkng J of Mth, ccepted 64 SS DRAGOMIR nd S WANG, A new inequlity of Ostrowski s type in L norm nd pplictions to some specil mens nd some numericl qudrture rules, Tmkng J of Mth, 8 997), SS DRAGOMIR nd S WANG, A new inequlity of Ostrowski s type in L p norm, Indin Journl of Mthemtics, 403) 998), SS DRAGOMIR nd S WANG, Applictions of Ostrowski s inequlity to the estimtion of error bounds for some specil mens nd some numericl qudrture rules, Appl Mth Lett, 998), AM FINK, A tretise on Grüss inequlity, submitted 68 G GRÜSS, Über ds Mximum des bsoluten Betrges von b b ) fx)dx gx)dx, Mth Z, ), DS Anlysis, Kluwer Acdemic Publishers, Dordrecht, 993 b b fx)gx)dx MITRINOVIĆ, JE PEČARIĆ nd AM FINK, Clssicl nd New Inequlities in

412 BIBLIOGRAPHY DS MITRINOVIĆ, JE PEČARIĆ nd AM FINK, Inequlities for Functions nd Their Integrls nd Derivtives, Kluwer Acdemic Publishers, Dordrecht, TC PEACHEY, A MCANDREW nd SS DRAGOMIR, The best constnt in n inequlity of Ostrowski type, Tmkng J of Mth, 303) 999), 9-7 J ROUMELIOTIS, PCERONE nd SS DRAGOMIR, An Ostrowski type inequlity for weighted mppings with bounded second derivtives, J KSIAM, 3) 999), M TORTORELLA, Closed Newton-Cotes qudrture rules for Stieltjes integrls nd numericl convolution of life distributions, Sim J Sci Stt Comput, 4) 990),