Hadamard-Type Inequalities for s Convex Functions I
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1 Punjb University Journl of Mthemtics ISSN 6-56) Vol. ). 5-6 Hdmrd-Tye Ineulities for s Convex Functions I S. Hussin Dertment of Mthemtics Institute Of Sce Technology, Ner Rwt Toll Plz Islmbd Highwy, Islmbd Emil: sbirhus@gmil.com M. I. Bhtti Dertment of Mthemtics University of Engineering nd Technology, Lhore, Pkistn Emil: iblbhtti@uet.edu.k M. Ibl Dertment of Mthemtics University of Engineering nd Technology, Lhore, Pkistn Emil: mibl.bki@gmil.com Abstrct.In this er we give refined uer bound for unit, or smller intervls nd refinement of Hermite Hdmrd Ineulity for s convex functions in second sense. We lso estblish severl Hdmrd tye Ineulities for differentible nd twice differentible functions bsed on concvity nd s convexity with lictions for some secil mens. AMS MOS) Subject Clssifiction Codes:]6D5, 6D Key Words: Hdmrd s ineulity; s convex functions; concve functions; Bet function.. INTRODUCTION Let f : I R R be convex ming defined on the intervl I of rel numbers nd, b I, with < b. The following double ineulity: ) + b f b f) + fb) fx)dx.) b = I R, is sid to be convex on I if ineulity ft x + t) y) t fx) + t) fy) holds for ll x, y I nd t, ]. Geometriclly, this mens tht if P, Q nd R re three distinct oints on grh of f with Q between P nd R, then Q is on or below chord PR. In the er 6], H. Hudzik nd L. Mligrnd considered, mong others, the clss of 5
2 5 S. Hussin, M. I. Bhtti nd M. Ibl functions which re s convex in the second sense. This clss is defined s follows: A function f :, ) R is sid to be s convex in the second sense if ft x + t) y) t s fx) + t) s fy).) holds for ll x, y, ), t, ] nd for some fixed s, ]. It my be noted tht every -convex function is convex. In the sme er 6] H. Hudzik nd L. Mligrnd discussed few results connecting with s convex functions in second sense nd some new results bout Hdmrd s ineulity for s convex functions re discussed in,, 8], while on the other hnd there re mny imortnt ineulities connecting with -convex convex) functions ], but one of these is the clssicl Hermite-Hdmrd ineulity defined by ] ) + b f b f) + fb) fx) dx b for, b] R. In 5], S. S. Drgomir et l. roved vrint of Hermite-Hdmrd s ineulity for s convex functions in second sense. Theorem. Suose tht f :, ), ) is s convex function in the second sense, where s, ], nd let, b, ), < b. If L, b], then the following ineulity holds ) + b s f b fx) dx b f) + fb)..3) s + The constnt k = is the best ossible in the second ineulity in.3). Their result ws imroved in 7], where Jgers gve both the uer nd lower bound for the constnt cs) in the ineulity ) + b b cs) f fx) dx. b He roved tht s + cs) s s s ) s s. s + In 3, ] S. S. Drgomir et l. discussed ineulities for differentible nd twice differentible functions connecting with the H-H Ineulity on the bsis of the following Lemms. Lemm. Let f : I R R be twice differentible function on I with f L, b], then f) + fb) b b ) fx) dx = t t) f t + t)b). b Lemm 3. Let f : I R R be differentible function on I,, b I with < b nd f L, b], then ) + b f b b ) fx) dx = t) f t + t) + b ) + b f tb + t) + b )].
3 Hdmrd-tye ineulities for s convex functions I We give here definition of Bet function of Euler tye which will be helful in our next discussion, which is for x, y > defined s βx +, y + ) = t x t) y. This er is orgnized s follows. After this Introduction, in section we discuss some s Hermite Hdmrd tye ineulities for differentible functions, in section 3 we give lictions of the results from section?? for secil mens nd in section we will discuss refinement of s Hermite Hdmrd ineulity nd its refined uer bound for unit, or smller, intervls.. INEQUALITIES FOR DIFFERENTIABLE FUNCTIONS Theorem. Let f : I R, I, ), be differentible function on I such tht f L, b], where, b I, < b. If f is s-convex on, b] for some fixed s, ] nd, then ) + b f b fx) dx b { f ) + s + ) f +b {s + )s + )} ) } + { f b) +s + ) f +b ) } {s + )s + )} = βs +, ) f ) + βs +, ) + b f βs +, ) f b) βs +, ) + + b f ) ) +.) ) ] ) Proof. By Lemm 3 ) + b f b fx) dx b b ) t) f t + t) + b ) + t) f t b + t) + b ) ].) f is s convex on, b] for t, ] f t + t) + b ) ) + b t s f ) + t) s f
4 5 S. Hussin, M. I. Bhtti nd M. Ibl Now t) f t + t) + b ) ) f ) t s t) + + b f t) +s ) = βs +, ) f ) + β, s + ) + b f = f ) + s + ) f ) +b s + )s + ) t) f t + t) + b ) = t) t) f t + t) + b ) By Hölder s Ineulity for > with = t) f t + t) + b ) t) f t + t) + b ) = t) f t + t) + b ) f ) + s + ) f ) +b ] s + )s + ) ) ) t) ) ) = f ) βs +, ) + β, s + ) + b ] f. Anlogously t) f t b + t) + b ) f b) + s + ) f ) +b ].) s + )s + ) ) = f b) βs +, ) + β, s + ) + b ] f By using.3) nd.) in.) we get.). Theorem 5. Let f : I R, I, ), be differentible function on I such tht f L, b], where, b I, < b. If f is concve on, b] for > with =, then ) + b f b ) )] b ) 3 + b + 3b fx) dx f + f b + ).5).3)
5 Hdmrd-tye ineulities for s convex functions I Proof. Similrly s in Theorem by using Hölder s Ineulity for > with = obtin t) f t + t) + b ) ) t) f t + t) + b ) = + ) f t + t) + b ) ) we ).6) f is concve on, b], by Integrl Jensen s Ineulity cf. 9]) we obtin f t + t) + b ) = t f t + t) + b ) t ) ) f t + t) +b ) t = f t + t) + b )) ) = 3 + b f..7) Anlogously f t b + t) + b ) ) + 3b f.8) By using.6).8) in.) we get.5). Theorem 6. Let f : I R, I, ), be differentible function on I such tht f L, b], where, b I, < b. If f is s convex on, b] for some fixed s, ] nd > with =, then ) + b f b fx) dx b ) b ) ) f ) + + b ) + ) s + f + ) ] f b) + + b ) f.9) ) b ) f ) + + b ) + ) f + ) ] f b) + + b ) f
6 56 S. Hussin, M. I. Bhtti nd M. Ibl Proof. We roceed similr to roof of Theorem 5. By s convexity of f we obtin f t + t) + b ) f ) + ) f +b..) s + Anlogously f t b + t) + b ) By using.),.) nd.6) in.) we get.9). And the second ineulity follows from the fcts ) s, ] nd > we hve. f b) + f ) +b..) s + Theorem 7. Let f : I R, I, ), be differentible function on I such tht f L, b], where, b I, < b. If f is s concve on, b] for some fixed s, ] nd > with =, then ) + b f b fx) dx b ) ) ] b ) s + ) 3 + b f + + 3b f.) Proof. we roceed similrly s in Theorem 6. By s concvity of f we obtin f t + t) + b ) ) 3 + b s f. Anlogously f t b + t) + b ) ) + 3b s f. Now.) is immedite from.). Vrints of these results for twice differentible functions re given below. These cn be roved in similr wy bsed on Lemm. Theorem 8. Let f : I R, I, ), be twice differentible function on I such tht f L, b], where, b I, < b. If f is s convex on, b] for some fixed s, ] nd, then f) + fb) b b fx) dx b ) 6 = b ) 6 f ) + f b) s + )s + 3) ]. βs +, ){ f ) + f b) }]. Theorem 9. Let f : I R, I, ), be twice differentible function on I such tht f L, b], where, b I,, b. If f is concve on, b] for > with = then f) + fb) b ) b ) fx) dx + b b f β +, + )].,
7 Hdmrd-tye ineulities for s convex functions I Theorem. Let f : I R, I, ), be twice differentible function on I such tht f L, b], where, b I, < b. If f is s convex on, b] for some fixed s, ] nd > with =, then f) + fb) b fx) dx b b ) f ) + f b) ] s + ) β +, + )]. Theorem. Let f : I R, I, ), be twice differentible function on I such tht f L, b], where, b I, < b. If f is s concve on, b] for some fixed s, ] nd > with =, then f) + fb) b ) fx) dx b s + b b ) f β +, + )]. Remrk. For s =, reltions.),.5),.9) nd.) rovide the right estimte of left clssicl Hdmrd difference, tht is, the new imrovements of left Hdmrd ineulity. Remrk 3. For s =, reltions in Theorems 8- rovide the right estimte of right clssicl Hdmrd difference, tht is, the new imrovements of right Hdmrd ineulity. 3. APPLICATIONS FOR SPECIAL MEANS Let us recll the following mens for two ositive numbers. ) The Arithmetic men A A, b) = + b,, b > ) The Hrmonic men H H, b) = b + + +)b ) b + b,, b > 3) The Logrithmic men, if = b;, b > L L, b) = ], if b. ) The Identric men, if = b;, b > I I, b) = ) b b b e, if b. 5) The Logrithmic { men, if = b;, b > L L, b) = b ln b ln, if b. The following ineulity is well known in the literture: H L I A. It is lso known tht L is monotoniclly incresing over R, denoting L = I nd L = L.
8 58 S. Hussin, M. I. Bhtti nd M. Ibl Proosition. Let >, < < b nd =. Then one hs the ineulity. H, b) L, b) b ) 6 3 b 3 A/ 3, b 3) 3.) Proof. By Theorem 8 lied for the ming fx) = x for s = we hve + ] b ln b ln b / ) + b 3 b 3, 6 / which is euivlent to 3.). Another result which is connected with Logrithmic men L, b) is the following one: Proosition 5. Let >, < < b nd =, then A, b ) L, b) )b ) A / ), b )) Proof. Follows by Theorem 8, setting fx) = x for s =. Another result which is connected with Logrithmic men L, b) is the following one: Proosition 6. Let >, < < b nd =, then A, b) 3 / { I, b) ex + A, b) ) / + b + A, b) ) }] / Proof. Follows by Theorem, setting fx) = ln x for s =. Remrk 7. By selecting some other convex functions, in the sme wy s bove, we cn find out some new reltions connecting to some secil mens.. REFINEMENT AND NEW REFINED UPPER BOUND FOR S-HERMITE HADAMARD INEQUALITY To find new refined uer bound we integrte.) w.r.t t over, b], ] x y where, x b+ b) y x + ) y fu) du b ) L s s, b) fx) + L s s, b) fy)], L α, β) = β+ α +, α β, >. β α) + ) For better right bound of Hermite Hdmrd Ineulity for s-convex function in second, we comre the bove bound with usul one, f)+fb). Suose the bove is less thn the usul uer bound, tht is, b fx) b) ) fy) fx)+fy) or, b ] fx) + ) b) ] fy) fx) + fy). Consider b = + λ for λ > such tht its cube nd higher owers roching zero. + λ) ] fx) + ) λ) ] fy) fx) + fy) So, ll we need, for the bove being true, is tht + λ) ; ) λ) ) i.e, + λ ] ) ] ; ) λ
9 Hdmrd-tye ineulities for s convex functions I By binomil exnsion, λ + s) λ ] ; ) λ s) ) λ ] s s λ + s λ s + ;, s )s λ + ) s λ s +.) From.) we get λ. This mens we hve imroved the uer bound of Hermite Hdmrd ineulity for s convex function in second sense, when the distnce between nd b is lmost one. The most interesting thing is tht ll linking work is with the intervl, ] better thn other. This discussion gives the following result. Theorem 8. Let f :, +λ] R be s convex function in second sense for < λ, < nd s, ], then ) + λ s f +λ ft) { s λ s + s λ + λ} s fx) + { s } ] )s λ + ) s λ fy). The following result is relted with the imrovement of ineulity.3). Theorem 9. Suose tht f :, ), ) is n s-convex function in the second sense, where s, ], nd let, b, ), < b. If f L, b], then f) + fb) b fx) dx s + b t s f) + t) s fb) ft + t) b).) nd b ) + b fx) dx s f b s s ft + t) b) + ft b + t) ) + ) + b f.3) Proof. From ineulity.) t s f) + t) s fb) ft + t) b) Integrting w.r.t t over,] f) + fb) s + = t s f) + t) s fb) ft + t) b) t s f) + t) s fb) ft + t) b). ft + t) b) t s f) + t) s fb) ft + t) b). { t s f) + t) s fb) ft + t) b) },
10 6 S. Hussin, M. I. Bhtti nd M. Ibl which is euivlent to.). Agin by definition fx) + fy) s f ) x + y fx) + fy) f s. ) ) x + y = fx) + fy) x + y s f ) fx) + fy) x + y s f By setting, x t + t) b nd y t b + t) for t, ], we hve ) ft + t) b) + ft b + t) ) + b s f ) ft + t) b) + ft b + t) ) + b s f. Integrting w.r.t t over,] ] ) + b s ft + t) b) + ft b + t) ) f { )} ft + t) b) + ft b + t) ) + b s f. From here we get.3). Acknowledgement : We thnk the creful referee nd Editor for vluble comments nd suggestions, which we hve used to imrove the finl version of this er REFERENCES ] M. Alomri nd M. Drus, On Co-ordinted s convex functions, Inter. Mth. Forum, 3) 8) ] M. Alomri nd M. Drus, Hdmrd-tye ineulities for s convex functions, Inter. Mth. Forum, 3) 8) ] S. S. Drgomir, On some ineulities for differentible convex functions nd lictions, submitted) ] S. S. Drgomir, C. E. M. Pierce, Selected Toics on Hermite-Hdmrd Ineulities nd Alictions. RGMIA, Monogrhs. 5] S. S. Drgomir, S. Fitztrick, The Hdmrd s ineulity for s convex functions in the second sense, Demonstrtio Mth. 3 ) 999) ] H. Hudzik, L. Mligrnd, Some remrks on s convex functions, Aeutiones mth. 8 99) -. 7] B. Jgers, On Hdmrd-tye ineulity for s convex functions. htt://wwwhome.cs.utwente.nl/ jgers/lhfrmes/alh.df. 8] U. S. Kirmci et l., Hdmrd-tye ineulities for s convex functions, Al. Mth. Com., 937), ] D. S. Mitrinović, J. E. Pečrić, A. M. Fink, Clssicl nd New Ineulities in Anlysis, Kluwer Acdemic Publishers, 993,. 6,, 5. ] J. Pečrić, F. Proschn, Y. L. Tong, Convex functions, rtil orderings nd sttisticl lictions, Acdemic Press, Inc., 99,. 37.
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