Sharp bounds for Sándor mean in terms of arithmetic, geometric and harmonic means


 Rosemary Beatrice Pope
 1 years ago
 Views:
Transcription
1 Qian t al. Journal of Inqualitis and Applications (015) 015:1 DOI /s R E S E A R C H Opn Accss Sharp bounds for Sándor man in trms of arithmtic, gomtric and harmonic mans WiMao Qian 1, YuMing Chu * and XiaoHui Zhang * Corrspondnc: School of Mathmatics and Computation Scincs, Hunan City Univrsity, Yiyang, 41000, China Full list of author information is availabl at th nd of th articl Abstract In th articl, w prsnt th bst possibl paramtrs α 1, α, β 1, β (0, 1) and α, α 4, β, β 4 (0, 1/) such that th doubl inqualitis α 1 A(a, b)+(1 α 1 )H(a, b)<x(a, b)<β 1 A(a, b)+(1 β 1 )H(a, b), α A(a, b)+(1 α )G(a, b)<x(a, b)<β A(a, b)+(1 β )G(a, b), H [ α a +(1 α )b, α b +(1 α )a ] < X(a, b)<h [ β a +(1 β )b, β b +(1 β )a ], G [ α 4 a +(1 α 4 )b, α 4 b +(1 α 4 )a ] < X(a, b)<g [ β 4 a +(1 β 4 )b, β 4 b +(1 β 4 )a ] hold for all a, b >0witha b.hr,x(a, b), A(a, b), G(a, b)andh(a, b)arthsándor, arithmtic, gomtric and harmonic mans of a andb, rspctivly. MSC: 6E60 Kywords: Sándor man; arithmtic man; gomtric man; harmonic man 1 Introduction Lt r R and a, b >0witha b. Thn th harmonic man H(a, b), gomtric man G(a, b), logarithmic man L(a, b), Siffrt man P(a, b), arithmtic man A(a, b), Sándor man X(a, b)[1]andrth powr man M r (a, b)ofa and b ar, rspctivly, dfind by and H(a, b)= ab a + b, G(a, b)= a b ab, L(a, b)= log a log b, (1.1) P(a, b)= a b arcsin( a b a+b ), + b G(a,b) A(a, b)=a, X(a, b)=a(a, b) P(a,b) 1 (1.) ( a r + b r ) 1/r M r (a, b)= (r 0), M 0 (a, b)= ab. (1.) It is wll known that M r (a, b) is continuous and strictly incrasing with rspct to r R for fid a, b >0witha b, and th inqualitis H(a, b)<g(a, b)<l(a, b)<p(a, b)<a(a, b) (1.4) hold for all a, b >0witha b. 015 Qian t al. This articl is distributd undr th trms of th Crativ Commons Attribution 4.0 Intrnational Licns (http://crativcommons.org/licnss/by/4.0/), which prmits unrstrictd us, distribution, and rproduction in any mdium, providd you giv appropriat crdit to th original author(s) and th sourc, provid a link to th Crativ Commons licns, and indicat if changs wr mad.
2 Qian t al. Journal of Inqualitis and Applications (015) 015:1 Pag of 1 Rcntly, th Sándor man has attractd th attntion of svral rsarchrs. In [], Sándor stablishd th inqualitis X(a, b)< P (a, b) A(a, b), A(a, b)g(a, b) A(a, b)p(a, b) < X(a, b)< P(a, b) P(a, b) G(a, b), A(a, b)l(a, b) X(a, b)> G(a,b) L(a,b) 1 A(a, b)[p(a, b)+g(a, b)], X(a, b)>, P(a, b) P(a, b) G(a, b) A [ ( (a, b)g(a, b) P(a, b)l(a, b) L(a,b) A(a,b) 1 1 < X(a, b)<a(a, b) ) ] G(a, b), P(a, b) [ ] A(a, b)+g(a, b) 4/ A(a, b)+g(a, b) P(a, b)<x(a, b)<a 1/ (a, b), P 1/(log π log ) (a, b)a 1 1/(log π log ) (a, b) [ ] A(a, b)+g(a, b) < X(a, b)<p 1 (a, b) for all a, b >0witha b. Yang t al. [] provd that th doubl inquality M p (a, b)<x(a, b)<m q (a, b) (1.5) holds for all a, b >0witha b if and only if p 1/ and q log /(1 + log )= In [4], Zhou t al. provd that th doubl inquality H α (a, b)<x(a, b)<h β (a, b) (1.6) holds for all a, b >0witha b if and only if α 1/ and β log /(1 + log )= , whr H p (a, b)=[(a p +(ab) p/ + b p )/] 1/p (p 0)andH 0 (p)= ab is th pth powrtyp Hronian man of a and b. Inqualitis (1.4) and(1.5) togthr with th idntitis H(a, b) =M 1 (a, b), G(a, b) = M 0 (a, b)anda(a, b)=m 1 (a, b)ladtothinqualitis H(a, b)<g(a, b)<x(a, b)<a(a, b) (1.7) for all a, b >0witha b. Lt a, b >0witha b, [0, 1/], f ()=H[a +(1 )b, b +(1 )a]andg()=g[a + (1 )b, b +(1 )a]. Thn both functions f and g ar continuous and strictly incrasing on [0, 1/]. Not that f (0) = H(a, b)<x(a, b)<f (1/) = A(a, b) (1.8) and g(0) = G(a, b) <X(a, b) <g(1/) = A(a, b). (1.9)
3 Qian t al. Journal of Inqualitis and Applications (015) 015:1 Pag of 1 Motivatd by inqualitis (1.7)(1.9), w naturally ask: what ar th bst possibl paramtrs α 1, α, β 1, β (0, 1) and α, α 4, β, β 4 (0, 1/) such that th doubl inqualitis α 1 A(a, b)+(1 α 1 )H(a, b)<x(a, b)<β 1 A(a, b)+(1 β 1 )H(a, b), α A(a, b)+(1 α )G(a, b)<x(a, b)<β A(a, b)+(1 β )G(a, b), H [ α a +(1 α )b, α b +(1 α )a ] < X(a, b)<h [ β a +(1 β )b, β b +(1 β )a ], G [ α 4 a +(1 α 4 )b, α 4 b +(1 α 4 )a ] < X(a, b)<g [ β 4 a +(1 β 4 )b, β 4 b +(1 β 4 )a ] hold for all a, b >0witha b? Th purpos of this papr is to answr this qustion. Lmmas In ordr to prov our main rsults, w nd four lmmas, which w prsnt in this sction. Lmma.1 Lt p (0, 1) and f ()= 1 [(1 p) +1] p +(1 p)(1 ) arcsin(). (.1) Thn th following statmnts ar tru: (1) If p =/, thn f ()<0for all (0, 1). () If p =1/, thn thr ists λ 1 (0, 1) such that f ()>0for (0, λ 1 ) and f ()<0for (λ 1,1). Proof Simpl computations lad to f (0) = 0, f (1) = π, (.) f ()= 1 [p +(1 p)(1 )] f 1(), (.) whr f 1 ()=(1 p) 4 (1 p)( p) + p. (.4) (1) If p =/,thn(.4)ladsto f 1 ()= ( 7 ) <0 (.5) 9 for (0, 1). Thrfor, f ()<0for (0, 1) follows asily from (.), (.)and(.5). () If p =1/,thn(.4)ladsto f 1 (0) = >0, f 1 (1) = 1 <0, (.6) f 1 ()=(1 p)[ (1 p) ( p) ] < ( 1 p ) <0 (.7) for (0, 1).
4 Qian t al. Journal of Inqualitis and Applications (015) 015:1 Pag 4 of 1 From (.6) and(.7) w clarly s that thr ists λ 0 (0, 1) such that f 1 () >0for (0, λ 0 )andf 1 ()<0for (λ 0,1). W divid th proof into two cass. Cas 1. (0, λ 0 ]. Thn f () > 0 follows asily from (.)and(.)togthrwithf 1 ()>0 on th intrval (0, λ 0 ). Cas. (λ 0,1).Thn(.) andf 1 ()<0onthintrval(λ 0,1) ladtothconclusion that f () is strictly dcrasing on [λ 0,1). From (.) andf (λ 0 ) > 0 togthr with th monotonicity of f () on[λ 0, 1) w clarly s that thr ists λ 1 (λ 0,1) (0, 1) such that f ()>0for (λ 0, λ 1 )andf ()<0for (λ 1,1). Lmma. Lt p (0, 1) and g()= p 1 +(1 p) (1 p) 1 + p arcsin(). (.8) Thn th following statmnts ar tru: (1) If p = 1/, thn g()>0 for all (0, 1). () If p =1/, thn thr ists μ 1 (0, 1) such that g()<0for (0, μ 1 ) and g()>0for (μ 1,1). Proof Simpl computations lad to g(0) = 0, g(1) = 1 p 1 π, (.9) g ()= 1 [p +(1 p) 1 ] g 1(), (.10) whr g 1 ()=p(p 1) 1 +1 p p. (.11) (1) If p = 1/, thn (.11)ladsto g 1 ()= 9( 1 1 ) >0 (.1) for (0, 1). Thrfor, g()>0 for all (0,1) follows asily from (.9), (.10)and(.1). () If p =1/,thn(.11)ladsto g 1 (0) = <0, g 1 (1) = 1 > 0, (.1) g 1 p(1 p) ()= >0 (.14) 1 for all (0, 1). From (.1) and(.14) w clarly s that thr ists μ 0 (0, 1) such that g 1 ()<0for (0, μ 0 )andg 1 ()>0for (μ 0,1).
5 Qian t al. Journal of Inqualitis and Applications (015) 015:1 Pag 5 of 1 W divid th proof into two cass. Cas 1. (0, μ 0 ]. Thn g()<0for (0, μ 0 ] follows asily from (.9) and(.10) togthr with g 1 ()<0onthintrval(0,μ 0 ). Cas. (μ 0,1).Thn(.10)andg 1 ()>0onthintrval(μ 0,1)ladtothconclusion that g() is strictly incrasing on [μ 0, 1). Not that g(μ 0 )<0, g(1) = 1 π > 0. (.15) From (.15) and th monotonicity of g()onthintrval[μ 0, 1) w clarly s that thr ists μ 1 (μ 0,1) (0, 1) such that g()<0for (μ 0, μ 1 )andg()>0for (μ 1,1). Lmma. Lt p (0, 1/) and h()=arcsin() 1 [1 + (1 p) ] 1 (1 p). (.16) Thn th following statmnts ar tru: (1) If p = 1/ /6= , thn h()>0for all (0, 1). () If p = 1/ 1 1//= , thn thr ists σ 1 (0, 1) such that h()<0for (0, σ 1 ) and h()>0for (σ 1,1). Proof Simpl computations lad to h(0) = 0, h(1) = π, (.17) h ()= 1 [1 (1 p) ] h 1(), (.18) whr h 1 ()= ( 16p 4 p +4p 8p +1 ) 4 + ( 16p 4 +p p +16p ) + ( 6p 6p +1 ). (.19) (1) If p = 1/ /6, thn (.19)ladsto h 1 ()= 4 9 ( 7 ) <0 (.0) for (0, 1). Thrfor, h() > 0 for all (0, 1) follows asily from (.17) and(.18) togthrwith (.10). () If p = 1/ 1 1//, thn h 1 (0) = ( 6p 6p +1 ) >0, h 1 (1) = 4p(1 p)<0, (.1) h 1 ()=4( 16p 4 p +4p 8p +1 ) + ( 16p 4 +p p +16p ). (.)
6 Qian t al. Journal of Inqualitis and Applications (015) 015:1 Pag 6 of 1 Not that 16p 4 p +4p 8p +1= >0, (.) 16p 4 p +16p 1 = < 0. (.4) It follows from (.)(.4)that h 1 () <4( 16p 4 p +4p 8p +1 ) + ( 16p 4 +p p +16p ) = ( 16p 4 p +16p 1 ) <0 (.5) for (0, 1). From (.1) and(.5) w clarly s that thr ists σ 0 (0, 1) such that h 1 ()>0for (0, σ 0 )andh 1 ()<0for (σ 0,1). W divid th proof into two cass. Cas 1. (0, σ 0 ]. Thn h()<0for (0, σ 0 ] follows asily from (.17) and(.18) togthr with h 1 ()>0onthintrval(0,σ 0 ). Cas. (σ 0,1).Thn(.18)andh 1 ()<0onthintrval(σ 0,1)ladtothconclusion that h() is strictly incrasing on (σ 0, 1). Thrfor, thr ists σ 1 (σ 0,1) (0, 1) such that h() <0for (σ 0, σ 1 )andh() >0for (σ 1, 1) follows from (.17) andh(σ 0 )<0 togthr with th monotonicity of h()onthintrval(σ 0,1). Lmma.4 Lt p (0, 1/) and J()=arcsin() 1 1 (1 p). (.6) Thn th following statmnts ar tru: (1) If p = 1/ 6/6= , thn J()>0for all (0, 1). () If p = 1/ 1 1/ /= , thn thr ists τ 1 (0, 1) such that J()<0for (0, τ 1 ) and h()>0for (τ 1,1). Proof Simpl computations lad to J(0) = 0, J(1) = π, (.7) J ()= 1 [1 (1 p) ] J 1(), (.8) whr J 1 ()= ( 16p 4 p +4p 8p +1 ) ( 1p 1p +1 ). (.9) (1) If p = 1/ 6/6, thn (.9)ladsto J 1 ()= 4 9 >0 (.0) for (0, 1).
7 Qian t al. Journal of Inqualitis and Applications (015) 015:1 Pag 7 of 1 Thrfor, J() > 0 for all (0, 1) follows asily from (.7) and(.8) togthrwith (.0). () If p = 1/ 1 1/ /, thn (.9)ladsto J 1 (0) = ( 1p 1p +1 ) <0, J 1 (1) = 4p ( 4p 8p +p +1 ) >0, (.1) J 1 ()=( 16p 4 p +4p 8p +1 ) >0 (.) for (0, 1). It follows from (.1) and(.) that thr ists τ 0 (0, 1) such that J 1 () <0for (0, τ 0 )andj 1 ()>0for (τ 0,1). W divid th proof into two cass. Cas 1. (0, τ 0 ]. Thn J()<0for (0, τ 0 ] follows asily from (.7) and(.8) togthr with J 1 ()<0onthintrval(0,τ 0 ). Cas. (τ 0,1).Thn(.8)andJ 1 ()>0onthintrval(τ 0,1)ladtothconclusion that J() is strictly incrasing on (τ 0,1). Thrfor, thr ists τ 1 (τ 0,1) (0, 1) such that J()<0for (τ 0, τ 1 )andj()>0 for (τ 1, 1) follows from (.7) andj(τ 0 ) < 0 togthr with th monotonicity of J() on th intrval (τ 0,1). Main rsults Thorm.1 Th doubl inquality α 1 A(a, b)+(1 α 1 )H(a, b)<x(a, b)<β 1 A(a, b)+(1 β 1 )H(a, b) holds for all a, b >0with a bifandonlyifα 1 1/ = and β 1 /. Proof Sinc H(a, b), X(a, b)anda(a, b) ar symmtric and homognous of dgr on, w assum that a > b >0.Lt =(a b)/(a + b) (0, 1) and p (0, 1). Thn (1.1)and(1.)lad to X(a, b) H(a, b) A(a, b) H(a, b) = 1 arcsin() 1 (1 ), (.1) Lt log X(a, b) pa(a, b)+(1 p)h(a, b) = 1 arcsin() 1 log [ p +(1 p) ( 1 )]. (.) 1 arcsin() F()= 1 log [ p +(1 p) ( 1 )]. (.) Thn simpl computations lad to F ( 0 +) =0, (.4) F(1) = log p 1, (.5) F ()= 1 f (), (.6) 1 whr f ()isdfindby(.1).
8 Qian t al. Journal of Inqualitis and Applications (015) 015:1 Pag 8 of 1 W divid th proof into two cass. Cas 1. p =/.Thn(.)(.4) and(.6) togthr with Lmma.1(1) lad to th conclusion that X(a, b)< A(a, b)+ 1 H(a, b). (.7) Cas. p =1/. Thn(.6) and Lmma.1() lad to th conclusion that thr ists λ 1 (0, 1) such that F() is strictly incrasing on (0, λ 1 ] and strictly dcrasing on [λ 1,1). Not that (.5)bcoms F(1) = 0. (.8) It follows from (.)(.4) and(.8) togthr with th picwis monotonicity of F() that X(a, b)> 1 ( A(a, b)+ 1 1 ) H(a, b). (.9) Not that 1 arcsin() 1 (1 ) lim = 0 +, (.10) lim 1 1 arcsin() 1 (1 ) = 1. (.11) Thrfor, Thorm.1 follows from (.7) and(.9) in conjunction with th following statmnts. If α 1 >/,thnquations(.1) and(.10) lad to th conclusion that thr ists δ 1 (0, 1) such that X(a, b)<α 1 A(a, b)+(1 α 1 )H(a, b) for all a > b >0with (a b)/(a + b) (0, δ 1 ). If β 1 <1/, thnquations(.1) and(.11) lad to th conclusion that thr ists δ (0, 1) such that X(a, b)>β 1 A(a, b)+(1 β 1 )H(a, b) for all a > b >0with (a b)/(a + b) (1 δ,1). Thorm. Th doubl inquality α A(a, b)+(1 α )G(a, b)<x(a, b)<β A(a, b)+(1 β )G(a, b) holds for all a, b >0with a bifandonlyifα 1/ and β 1/ = Proof Sinc A(a, b), G(a, b)andx(a, b) ar symmtric and homognous of dgr on, w assum that a > b >0.Lt =(a b)/(a + b) (0, 1) and p (0, 1). Thn (1.1)and(1.)lad to X(a, b) G(a, b) A(a, b) G(a, b) = 1 arcsin() , (.1)
9 Qian t al. Journal of Inqualitis and Applications (015) 015:1 Pag 9 of 1 Lt log X(a, b) pa(a, b)+(1 p)g(a, b) = 1 arcsin() 1 log [ p +(1 p) 1 ]. (.1) 1 arcsin() G()= 1 log [ p +(1 p) 1 ]. (.14) Thn simpl computations lad to G ( 0 +) = 0, (.15) G(1) = log p 1, (.16) G ()= 1 g(), (.17) 1 whr g()isdfindby(.8). W divid th proof into two cass. Cas 1. p = 1/. Thn (.1)(.15) and(.17) togthr with Lmma.(1) lad to th conclusion that X(a, b)> 1 A(a, b)+ G(a, b). (.18) Cas. p =1/. Thn from Lmma.() and (.17) w know that thr ists μ 1 (0, 1) such that G() is strictly dcrasing on (0, μ 1 ] and strictly incrasing on [μ 1, 1). Not that (.16)bcoms G(1) = 0. (.19) It follows from (.1)(.15)and(.19) togthr with th picwis monotonicity of G() that X(a, b)< 1 ( A(a, b)+ 1 1 ) G(a, b). (.0) Not that 1 arcsin() 1 1 lim = 1 1, (.1) 1 arcsin() 1 1 lim 1 1 = 1 1. (.) Thrfor, Thorm. follows asily from (.1) and(.18) togthrwith(.0) (.). Thorm. Lt α, β (0, 1/). Thn th doubl inquality H [ α a +(1 α )b, α b +(1 α )a ] < X(a, b)<h [ β a +(1 β )b, β b +(1 β )a ]
10 Qian t al. Journal of Inqualitis and Applications (015) 015:1 Pag 10 of 1 holds for all a, b >0with a bifandonlyifα 1/ 1 1//= and β 1/ /6= Proof Sinc H(a, b)andx(a, b) ar symmtric and homognous of dgr on, w assum that a > b >0.Lt =(a b)/(a + b) (0, 1) and p (0, 1/). Thn (1.1)and(1.)ladto Lt log H[pa +(1 p)b, pb +(1 p)a] X(a, b) = log [ 1 (1 p) ] 1 arcsin() +1. (.) H()=log [ 1 (1 p) ] 1 arcsin() +1. (.4) Thn simpl computations lad to H ( 0 +) =0, (.5) H(1) = 1 + log +log ( p p ), (.6) H ()= 1 h(), (.7) 1 whr h()isdfindby(.16). W divid th proof into four cass. Cas 1. p = 1/ /6. Thn (.)(.5) and(.7) togthr with Lmma.(1) lad to [( 1 X(a, b) <H 6 ) ( 1 a ) ( 1 b, 6 ) ( 1 b ) ] a. Cas. 0 < p < 1/ /6. Lt q =(1 p) and 0 +, thn 1/ < q < 1 and powr sris pansion lads to ( H()= q 1 ) + o ( ). (.8) Equations (.), (.4)and(.8) lad to th conclusion that thr ists 0 < δ <1such that X(a, b)>h [ pa +(1 p)b, pb +(1 p)a ] (.9) for all a > b >0with(a b)/(a + b) (0, δ). Cas. p = 1/ 1 1//. Thn (.7) and Lmma.() lad to th conclusion that thr ists σ 1 (0, 1) such that H() is strictly dcrasing on (0, σ 1 ] and strictly incrasing on [σ 1,1). Not that (.6)bcoms H(1) = 0. (.0)
11 Qian t al. Journal of Inqualitis and Applications (015) 015:1 Pag 11 of 1 Thrfor, [( 1 X(a, b) >H 1 1 ) ( 1 a ) ( 1 b, 1 1 ) ( 1 b follows from (.)(.5)and(.0) togthr with th picwis monotonicity of H(). Cas 4. 1/ 1 1// < p < 1/. Thn (.6)ladsto ) ] a H(1) > 0. (.1) Equations (.) and(.4) togthr with inquality (.1) imply that thr ists 0 < δ <1suchthat X(a, b)<h [ pa +(1 p)b, pb +(1 p)a ] for a > b >0with(a b)(a + b) (1 δ,1). Thorm.4 Lt α 4, β 4 (0, 1/). Thn th doubl inquality G [ α 4 a +(1 α 4 )b, α 4 b +(1 α 4 )a ] < X(a, b)<g [ β 4 a +(1 β 4 )b, β 4 b +(1 β 4 )a ] holds for all a, b >0with a bifandonlyifα 4 1/ 1 1/ /= and β 4 1/ 6/6= Proof Sinc G(a, b)andx(a, b) ar symmtric and homognous of dgr on, w assum that a > b >0.Lt =(a b)/(a + b) (0, 1) and p (0, 1/). Thn (1.1)and(1.)ladto G[pa +(1 p)b, pb +(1 p)a] log X(a, b) = 1 log[ 1 (1 p) ] 1 arcsin() +1. (.) Lt K()= 1 log[ 1 (1 p) ] 1 arcsin() +1. (.) Thn simpl computations lad to K ( 0 +) =0, (.4) K(1) = 1 + log + 1 log( p p ), (.5) K ()= 1 J(), (.6) 1 whr J()isdfindby(.6). W divid th proof into four cass.
12 Qian t al. Journal of Inqualitis and Applications (015) 015:1 Pag 1 of 1 Cas 1. p = p 0 = 1/ 6/6. Thn X(a, b)<g [ p 0 a +(1 p 0 )b, p 0 b +(1 p 0 )a ] follows from (.)(.4)and(.6) togthr with Lmma.4(1). Cas. 0 < p < 1/ 6/6. Lt q =(1 p) and 0 +, thn / < q < 1 and powr sris pansion lads to K()= 1 ( q ) + o ( ). (.7) From (.), (.)and(.7) w clarly s that thr ists 0 < δ <1suchthat X(a, b)>g [ pa +(1 p)b, pb +(1 p)a ] for a > b >0with(a b)/(a + b) (0, δ). Cas. p = p 1 = 1/ 1 1/ /. Thn (.6) and Lmma.4() lad to th conclusion that thr ists τ 1 (0, 1) such that K() isstrictlydcrasingon(0,τ 1 ]andstrictly incrasing on [τ 1,1). Not that (.5)bcoms K(1) = 0. (.8) Thrfor, X(a, b)>g [ p 1 a +(1 p 1 )b, p 1 b +(1 p 1 )a ] follows from (.)(.4)and(.8) togthr with th picwis monotonicity of K(). Cas 4. 1/ 1 1/ / < p < 1/. Thn (.5)ladsto K(1) > 0. (.9) Equations (.) and(.) togthr with inquality (.9) imply that thr ists 0 < δ <1suchthat X(a, b)<g [ pa +(1 p)b, pb +(1 p)a ] (.40) for a > b >0with(a b)/(a + b) (1 δ,1). Compting intrsts Th authors dclar that thy hav no compting intrsts. Authors contributions All authors contributd qually to th writing of this papr. All authors rad and approvd th final manuscript. Author dtails 1 School of Distanc Education, Huzhou Broadcast and TV Univrsity, Huzhou, 1000, China. School of Mathmatics and Computation Scincs, Hunan City Univrsity, Yiyang, 41000, China.
13 Qian t al. Journal of Inqualitis and Applications (015) 015:1 Pag 1 of 1 Acknowldgmnts Th rsarch was supportd by th Natural Scinc Foundation of China undr Grants , and , and th Natural Scinc Foundation of Zhjiang Provinc undr Grant LY1A Rcivd: Fbruary 015 Accptd: Jun 015 Rfrncs 1. Sándor, J: Two sharp inqualitis for trigonomtric and hyprbolic functions. Math. Inqual. Appl. 15(), (01). Sándor, J: On two nw mans of two variabls. Nots Numbr Thory Discrt Math. 0(1), 19 (014). Yang, ZH, Wu, LM, Chu, YM: Sharp powr man bounds for Sándor man. Abstr. Appl. Anal. 014, Articl ID (014) 4. Zhou, SS, Qian, WM, Chu, YM, Zhang, XH: Sharp powrtyp Hronian man bounds for th Sándor and Yang mans. J. Inqual. Appl. 015, Articl ID 159 (015)
Quantum Graphs I. Some Basic Structures
Quantum Graphs I. Som Basic Structurs Ptr Kuchmnt Dpartmnt of Mathmatics Txas A& M Univrsity Collg Station, TX, USA 1 Introduction W us th nam quantum graph for a graph considrd as a ondimnsional singular
More informationType Inference and Optimisation for an Impure World.
Typ Infrnc and Optimisation for an Impur World. Bn Lippmir DRAFT Jun, 2009 A thsis submittd for th dgr of Doctor of Philosophy of th Australian National Univrsity 2 Dclaration Th work in this thsis is
More informationOptimization design of structures subjected to transient loads using first and second derivatives of dynamic displacement and stress
Shock and Vibration 9 (202) 445 46 445 DOI 0.3233/SAV2020685 IOS Prss Optimization dsign of structurs subjctd to transint loads using first and scond drivativs of dynamic displacmnt and strss Qimao Liu
More informationPedro Linares, Francisco Javier Santos, Mariano Ventosa, Luis Lapiedra
Masuring th impact of th uropan carbon trading dirctiv and th prmit assignmnt mthods on th Spanish lctricity sctor Pdro Linars, Francisco Javir Santos, Mariano Vntosa, Luis Lapidra Instituto d Invstigación
More informationIMPROVE CUSTOMERS LOYALTY IN ONLINE GAMING: AN EMPIRICAL STUDY Fan Zhao
Fan Zhao ABSTRACT In th past dcad, onlin gams hav bcom an important lctronic commrc application A good undrstanding of customr onlin gam bhaviors is critical for both rsarchrs and practitionrs, such as
More informationThe author(s) shown below used Federal funds provided by the U.S. Department of Justice and prepared the following final report:
Th author(s) shown blow usd Fdral funds providd by th U.S. Dpartmnt of Justic and prpard th following final rport: Documnt Titl: Author(s): Impact Munitions Data Bas of Us and Effcts Kn Hubbs ; David Klingr
More informationSuggestion of Promising Result Types for XML Keyword Search
Suggstion o Promising Rsult Typs or XML Kyword Sarch Jianxin Li, Chngi Liu and Rui Zhou Swinurn Univrsity o Tchnology Mlourn, Australia {jianxinli, cliu, rzhou}@swin.du.au Wi Wang Univrsity o Nw South
More informationStrong convergence theorems of Cesàrotype means for nonexpansive mapping in CAT(0) space
Tang et al. Fixed Point Theory and Applications 205 205:00 DOI 0.86/s3663050353y R E S E A R C H Open Access Strong convergence theorems of Cesàrotype means for nonexpansive mapping in CAT0 space Jinfang
More informationSolutions to Homework 6 Mathematics 503 Foundations of Mathematics Spring 2014
Solutions to Homework 6 Mathematics 503 Foundations of Mathematics Spring 2014 3.4: 1. If m is any integer, then m(m + 1) = m 2 + m is the product of m and its successor. That it to say, m 2 + m is the
More informationEffect of Electrical Stimulation on Bacterial Growth
Effct of Elctrical Stimulation on Bactrial Growth By Jrrold Ptrofsky Ph D Michal Laymon DPTSc* Wndy Chung DPTSc Klly Collins BS* TinNing Yang BS* Dpts. of Physical Thrapy Loma Linda Univrsity Loma Linda,
More informationWhat are the most abundant proteins in a cell?
What ar th most abundant s in a cll? Evn aftr rading svral txtbooks on s, on may still b lft wondring which of ths critical molcular playrs in th lif of a cll ar th most quantitativly abundant. Though
More informationCreate Change. 2015 Create Change Design Forums. 2015 Create Change Design Forums to register: www.accdchina.com
Crat Chang 2015 Crat Chang Dsign Forums 2015 Crat Chang Dsign Forums to rgistr: www.accdchina.com About Art Cntr Collg of Dsign Art Cntr Collg of Dsign, locatd in Pasadna, California, Unitd Stats, is on
More informationCoupled best proximity point theorems for proximally gmeirkeelertypemappings in partially ordered metric spaces
Abkar et al. Fixe Point Theory an Applications 2015) 2015:107 DOI 10.1186/s1366301503559 R E S E A R C H Open Access Couple best proximity point theorems for proximally gmeirkeelertypemappings in
More informationINTRODUCTORY SET THEORY
M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H1088 Budapest, Múzeum krt. 68. CONTENTS 1. SETS Set, equal sets, subset,
More informationNONLINEAR CONTROL OF WHEELED MOBILE ROBOTS
NONLINEAR CONTROL OF WHEELED MOBILE ROBOTS Maria Isabl Ribiro Pdro Lima mir@isr.ist.tl.pt pal@isr.ist.tl.pt Institto Sprior Técnico (IST) Institto d Sistmas Robótica (ISR) A.Roisco Pais, 49 Lisboa PORTUGAL
More informationProperties of BMO functions whose reciprocals are also BMO
Properties of BMO functions whose reciprocals are also BMO R. L. Johnson and C. J. Neugebauer The main result says that a nonnegative BMOfunction w, whose reciprocal is also in BMO, belongs to p> A p,and
More informationImportant result on the first passage time and its integral functional for a certain diffusion process
Lcturs Mtmátics Volumn 22 (21), págins 5 9 Importnt rsult on th first pssg tim nd its intgrl functionl for crtin diffusion procss Yousf ALZlzlh nd Bsl M. ALEidh Kuwit Univrsity, Kuwit Abstrct. In this
More informationShould I Stay or Should I Go? Migration under Uncertainty: A New Approach
Should Stay o Should Go? Migation und Unctainty: A Nw Appoach by Yasmn Khwaja * ctob 000 * patmnt of Economics, School of intal and Afican Studis, Unisity of ondon, Thonhaugh Stt, Russll Squa, ondon WC
More informationCategory 11: Use of Sold Products
11 Catgory 11: Us of Sold Products Catgory dscription T his catgory includs missions from th us of goods and srvics sold by th rporting company in th rporting yar. A rporting company s scop 3 missions
More informationIRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction
IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducible
More informationOn contractive cyclic fuzzy maps in metric spaces and some related results on fuzzy best proximity points and fuzzy fixed points
De la Sen et al. Fixed Point Theory and Applications (2015 2015:103 DOI 10.1186/s1366301503501 R E S E A R C H Open Access On contractive cyclic fuzzy maps in metric spaces and some related results
More informationMaintain Your F5 Solution with Fast, Reliable Support
F5 SERVICES TECHNICAL SUPPORT SERVICES DATASHEET Maintain Your F5 Solution with Fast, Rliabl Support In a world whr chang is th only constant, you rly on your F5 tchnology to dlivr no mattr what turns
More information( 1)2 + 2 2 + 2 2 = 9 = 3 We would like to make the length 6. The only vectors in the same direction as v are those
1.(6pts) Which of the following vectors has the same direction as v 1,, but has length 6? (a), 4, 4 (b),, (c) 4,, 4 (d), 4, 4 (e) 0, 6, 0 The length of v is given by ( 1) + + 9 3 We would like to make
More informationSOLUTIONS FOR PROBLEM SET 2
SOLUTIONS FOR PROBLEM SET 2 A: There exist primes p such that p+6k is also prime for k = 1,2 and 3. One such prime is p = 11. Another such prime is p = 41. Prove that there exists exactly one prime p such
More informationQ D = 100  (5)(5) = 75 Q S = 50 + (5)(5) = 75.
4. The rent control agency of New York City has found that aggregate demand is Q D = 1005P. Quantity is measured in tens of thousands of apartments. Price, the average monthly rental rate, is measured
More informationON EMLIKE ALGORITHMS FOR MINIMUM DISTANCE ESTIMATION. P.P.B. Eggermont and V.N. LaRiccia University of Delaware
March 1998 ON EMLIKE ALGORITHMS FOR MINIMUM DISTANCE ESTIMATION PPB Eggermont and VN LaRiccia University of Delaware Abstract We study minimum distance estimation problems related to maximum likelihood
More informationBasic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011
Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely
More informationSystems of First Order Linear Differential Equations
Sysms of Firs Ordr Linr Diffrnil Equions W will now urn our nion o solving sysms of simulnous homognous firs ordr linr diffrnil quions Th soluions of such sysms rquir much linr lgbr (Mh Bu sinc i is no
More informationU.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra
U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory
More informationGROUPS ACTING ON A SET
GROUPS ACTING ON A SET MATH 435 SPRING 2012 NOTES FROM FEBRUARY 27TH, 2012 1. Left group actions Definition 1.1. Suppose that G is a group and S is a set. A left (group) action of G on S is a rule for
More information