Ostrowski Type Inequalities and Applications in Numerical Integration. Edited By: Sever S. Dragomir. and. Themistocles M. Rassias

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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: 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:

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