ISRN Applied Mthemtics, Article ID 8958, 4 pges http://dx.doi.org/.55/4/8958 Reserch Article On Hermite-Hdmrd Type Inequlities for Functions Whose Second Derivtives Absolute Vlues Are s-convex Feixing Chen nd Xuefei Liu School of Mthemtics nd Sttistics, Chongqing Three Gorges University, Wnzhou, Chongqing 44, Chin Correspondence should be ddressed to Xuefei Liu; 4553@qq.com Received 5 September 3; Accepted 6 November 3; Published 5 Februry 4 Acdemic Editors: Y.-D. Kwon, Y. Wng, X.-S. Yng, nd W. Yeih Copyright 4 F. Chen nd X. Liu. This is n open ccess rticle distributed under the Cretive Commons Attribution License, which permits unrestricted use, distribution, nd reproduction in ny medium, provided the originl work is properly cited. Some new Hermite-Hdmrd type inequlities re obtined for functions whose second derivtives bsolute vlues re s-convex.. Introduction Let f : I R R be convex function on the intervl I; then for ny, b I with =bwe hve the following double inequlity: f +b b b. This remrkble result is well known in the literture s the Hermite-Hdmrdinequlity.Notethtsomeoftheclssicl inequlitiesfor mens cn be derived from for pproprite prticulr selections of the mpping f.bothinequlitieshold in the reversed direction if f is concve. Some refinements of the Hermite-Hdmrd inequlity on convex functions hve been extensively investigted by number of uthors e.g., 9. Definition see. A function f : I ; ; is sid to be s-convex on I,if the inequlity fαx+ α y α s f x + α s fy holds for ll x, y I with α,nd for some fixed s,. Itcnbeesilyseenthtfors=, s-convexity reduces to ordinry convexity of functions defined on,. In, Drgomir nd Fitzptrick proved vrint of Hermite-Hdmrd inequlity which holds for the s-convex functions. Theorem see. Suppose tht f : I,, is n s-convex function in the second sense, where s, nd let, b,, <b.iff L,, then the following inequlity holds: s f +b b b. 3 s+ The constnt k = /s+ isthebestpossibleinthesecond inequlity in 3. Definition 3 see 4. Wesy thtf:i R Ris P-convex function or tht f belongs to the clss PI,iff is nonnegtive nd for ll x, y I nd α, one hs fαx+ α y f x +fy. 4 Remrk 4. Applying Definition for s +, we get Definition 3. Along this pper we consider rel intervl I R,ndwe denote tht I is the interior of I. In, Brni et l. introduced the following theorems for twice differentible P-convex functions. Theorem 5 see. Let f:i Rbe twice differentible function on I such tht f is P-convex function on I.
ISRN Applied Mthemtics Suppose tht, b I with <bnd f L,b.Then, the following inequlity holds: b 4 b b f + f +b + f b. 5 Theorem 6 see. Let f:i Rbe twice differentible function on I. Assume tht p R, p>such tht f p/p is P-convex function on I.Supposetht, b I with <b nd f L,b.Then,thefollowinginequlityholds: b 6 b b /p πγp + Γp + 3/ f q + f +b q /q + f q b + f +b q /q. Theorem 7 see. Let f:i Rbe twice differentible function on I. Assume tht q such tht f q is P-convex function on I. Supposetht, b I with <bnd f L, b.then,thefollowinginequlityholds: b b b 4 f q + f +b q /q + f q b + f +b q /q. For recent results nd generliztions concerning Hermite-Hdmrd s inequlity for twice differentible functions see, 4 nd the references given therein. In this pper, we estblish some new inequlities of Hdmrd s type for the clss of s-convex functions in the second sense. 6 7. The New Hermite-Hdmrd Type Inequlities Lemm 8 see 3. Let f : I R R be twice differentible mpping on I,where, b I with <b.iff L, b, then the following equlity holds: = b 6 b b t f +t + t b +f t ++t b dt. 8 Theorem 9. Let f:i R Rbe differentible mpping on I,suchthtf L,b,where, b I with <b.if f is s-convex on, b for some fixed s, ; then the following inequlity holds: b b b f + f b + s+4 f +b/ 8 s+s+3 s+s+3 Proof. From Lemm 8,wehve b 6 b b. 9 t f +t + t b + f t ++t b dt. Becuse f is s-convex, we hve b 6 t = b 8 b b t s f + ts f +b +ts f b dt f + f b s+s+3 which completes the proof. + s+4 f +b/, s+s+3
ISRN Applied Mthemtics 3 The corresponding version for powers of the bsolute vlue of the second derivtive is incorported in the following theorems. Theorem. Let f:i R Rbe differentible mpping on I,suchthtf L,b,where, b I with <b.if f p/p is s-convex on, b for some fixed s, nd p>, then the following inequlity holds: b b = b 6 /p πγp + Γp + 3/ f q + f q /q +b/ s+ + f q b + f q /q +b/. s+ 3 b 6 /p πγp + Γp + 3/ f q + f q /q +b/ s+ + f q b + f q /q +b/, s+ where /p + /q =. We note tht the Bet nd Gmm functions re defined, respectively, s follows: Γ x = e x t x dt, βx,y= t y t x dt, βx,y= Γ x Γy Γx+y. x>, x>, y>, 4 Proof. From Lemm 8 ndusingthehölder inequlity, we hve b 6 b 6 b b t f +t + t b + f t ++t b dt t p /p dt f +t q /q + t b dt b 6 + f t q /q ++t b dt t p /p dt t s f q + t s f +b q /q dt + t s f q b + t s f +b q /q dt Then nd we get the desired result. t p πγp + dt = Γp + 3/, 5 Theorem. Let f:i R Rbe differentible mpping on I,suchthtf L,b,where, b I with <b.if f q is s-convex on, b for some fixed s, nd q, then the following inequlity holds: b b b /q 6 3 f q s+s+3 + s+4 f q +b/ s+s+3 + f q b f q +b / s+s+3 + s+4 s+s+3 /q /q. 6
4 ISRN Applied Mthemtics Proof. From Lemm 8 ndusingthewell-knownpowermen inequlity, we hve b 6 b b t f +t + t b + f t ++t b dt b 6 /q t dt t f +t q /q + t b dt + t f t q /q ++t b dt b /q 6 3 t t s f q + t t s f q b = b /q 6 3 + t s f +b q /q dt + t s f +b q /q dt f q s+s+3 + s+4 f q + b/ s+s+3 + f q b f q +b/ s+s+3 + s+4 s+s+3 This proves the theorem. /q /q. 7 Remrk. Applying Theorem 9 for s +,wegettheresult of Theorem 5. Remrk 3. Applying Theorem for s +,wegetthe result of Theorem 6. Remrk 4. Applying Theorem for s +,wegetthe result in Theorem 7. Conflict of Interests The uthors declre tht there is no conflict of interests regrding the publiction of this pper. Acknowledgment This work is supported by Youth Project of Chongqing Three Gorges University of Chin No. 3QN. References G. Frid, S. Abrmovich, nd J. Pečrić, More bout hermitehdmrd inequlities, cuchy s mens, nd superqudrcity, Journl of Inequlities nd Applictions, vol.,articleid 467,. M.W.Alomri,M.Drus,ndU.S.Kirmci, Someinequlities of Hermite-Hdmrd type for s-convex functions, Act Mthemtic Scienti, vol. 3, no. 4, pp. 643 65,. 3 M. Bessenyei nd Z. Páles, Hdmrd-type inequlities for generlized convex functions, Mthemticl Inequlities nd Applictions,vol.6,no.3,pp.379 39,3. 4 S. S. Drgomir, J. Pecric, nd Persson, Some inequlities of Hdmrd type, Soochow Journl of Mthemtics, no., pp. 335 34, 995. 5 A. El Frissi, Simple proof nd refinement of Hermite- Hdmrd inequlity, Journl of Mthemticl Inequlities,vol. 4, no. 3, pp. 365 369,. 6 X. Go, A note on the Hermite-Hdmrd inequlity, Journl of Mthemticl Inequlities,vol.4,no.4,pp.587 59,. 7 H.Kvurmci,M.Avci,ndM.E.Özdemir, New inequlities of Hermite-Hdmrd type for convex functions with pplictions, Journl of Inequlities nd Applictions, vol.,p.86,. 8 U.S.Kirmci,M.Klričić Bkul, M.E. Özdemir, nd J. Pečrić, Hdmrd-type inequlities for s-convex functions, Applied Mthemtics nd Computtion,vol.93,no.,pp.6 35,7. 9 C.E.M.PercendJ.Pečcrić, Inequlities for differentible mppings with ppliction to specil mens nd qudrture formul, Applied Mthemtics Letters, vol., no. 3, pp. 5 55,. M. Alomri, M. Drus, nd S. S. Drgomir, New inequlities of Hermite-Hdmrd type for functions whose second derivtives bsolute vlues re qusi-convex, Tmkng Journl of Mthemtics, vol. 4, no. 4, pp. 353 359,. S. S. Drgomir nd S. Fitzptrick, The Hdmrd inequlities for s-convex functions in the second sense, Demonstrtion Mthemtics,vol.3,no.4,pp.687 696,999. A. Brni, S. Brni, nd S. S. Drgomir, Refinements of Hermite-Hdmrd inequlities for functions when power of thebsolutevlueofthesecondderivtiveisp-convex, Journl of Applied Mthemtics, vol.,articleid65737,pges,. 3 A. Brni, S. Brni, nd S. S. Drgomir, Refinements of Hermite-Hdmrd type inequlity for functions whose second derivtives bsolute vlues re qusiconvex, RGMIA Reserch Report Collection, vol. 4, Article ID 69,. 4 S. Hussin, M. I. Bhtti, nd M. Iqbl, Hdmrd-type inequlities for s-convex functions I, Punjb University Journl of Mthemtics,vol.4,pp.5 6,9.
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