Research Article Hermite-Hadamard-Type Inequalities for r-preinvex Functions

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1 Hindwi Publishing Copotion Jounl of Applied Mthemtics Volume 3, Aticle ID 6457, 5 pges Resech Aticle Hemite-Hdmd-Type Inequlities fo -Peinvex Functions Wsim Ul-Hq nd Jved Iqbl Deptment of Mthemtics, Abdul Wli Khn Univesity Mdn, Mdn 3, Pkistn Coespondence should be ddessed to Wsim Ul-Hq; wsim474@hotmil.com Received 7 My 3; Revised 4 August 3; Accepted 4 August 3 Acdemic Edito: Zhongxio Ji Copyight 3 W. Ul-Hq nd J. Iqbl. This is n open ccess ticle distibuted unde the Cetive Commons Attibution License, which pemits unesticted use, distibution, nd epoduction in ny medium, povided the oiginl wok is popely cited. We im to find Hemite-Hdmd inequlity fo -peinvex functions. Also, it is investigted fo the poduct of n -peinvex function nd s-peinvex function.. Intoduction Sevel inteesting geneliztions nd extensions of clssicl convexity hve been studied nd investigted in ecent yes. Hnson [] intoduced the invex functions s geneliztion of convex functions. Lte, subsequent woks inspied fom Hnson s esult hve getly found the ole nd pplictions of invexity in nonline optimiztion nd othe bnches of pue nd pplied sciences. The bsic popeties nd ole of peinvex functions in optimiztion, equilibium poblems nd vitionl inequlities wee studied by Noo [, 3] nd Wei nd Mond [4]. The well known Hemite-Hdmd inequlity hs been extensively investigted fo the convex functions nd thei vint foms. Noo [5] nd Pchptte[6] studied these inequlities nd elted types fo peinvex nd log-peinvex functions. The Hemite-Hdmd inequlity ws investigted in [7] ton-convexpositivefunctionwhichisdefinedon n intevl [, b]. Cetin efinements of the Hdmd inequlity fo -convex functions wee studied in [8, 9]. Recently, Zbndn et l. [] extended nd efined the wok of [8]. Bessenyei [] studied Hemite-Hdmd-type inequlities fo genelized 3-convex functions. In this ppe, we intoduce the -peinvex functions nd estblish Hemite- Hdmd inequlity fo such functions by using the method of [].. Peliminies Let f:k R nd η(, ) : K K R be continuous functions, whee K R n is nonempty closed set. We use the nottions,, nd, fo inne poduct nd nom, espectively. We equie the following well known concepts nd esults which e essentil in ou investigtions. Definition (see [, 4]). Let u K.Then,thesetK is sid to be invex t u Kwith espect to η(, ) if u+tη(v,u) K, u,v K, t [, ]. () K is sid to be invex set with espect to η(, ) if it is invex t evey u K.TheinvexsetK is lso clled η connected set. Geometiclly, Definition sys tht thee is pth stting fom the point u which is contined in K.Thepoint V should not be one of the end points of the pth in genel, see []. This obsevtion plys key ole in ou study. If we equie tht V should be n end point of the pth fo evey pi of points u, V K,thenη(V,u) = V u,ndconsequently, invexity educes to convexity. Thus, evey convex set is lso n invex set with espect to η(v,u) = V u,buttheconvese is not necessily tue; see [4, 3] nd the efeences theein. Fo the ske of simplicity, we ssume tht K=[,+η(b,)], unless othewise specified. Definition (see [4]). The function f on the invex set K is sid to be peinvex with espect to η if f(u+tη(v,u)) ( t) f (u) +tf(v), u, V K, t [, ]. Note tht evey convex function is peinvex function, but the convese is not tue. Fo exmple, the function f(u) = ()

2 Jounl of Applied Mthemtics u is not convex function, but it is peinvex function with espect to η,whee η (V,u) ={ V u, u V, if V, uo V, u, othewise. The concepts of the invex nd peinvex functions hve plyed vey impotnt oles in the development of genelized convexpogmming, see [4 7]. Fo moe chcteiztions nd pplictions of invex nd peinvex functions, we efe to [3, 5, 8 5]. Antczk [6, 7] intoduced nd studied the concept of -invex nd -peinvex functions. Hee, we define the following. Definition 3. Apositivefunctionf on the invex set K is sid to be -peinvex with espect to η if, fo ech u, V K, t [, ]: f(u+tη(v,u)) { (( t) f (u) +tf (V)) /, = (f (u)) t (f (V)) t, =. (4) Note tht -peinvex functions e logithmic peinvex nd -peinvex functions e clssicl peinvex functions. It should be noted tht if f is -peinvex function, then f is peinvex function ( >). The well known Hemite-Hdmd inequlity fo convex function defined on the intevl [, b] is given by f ( +b ) b b f (x) dx (3) f () +f ; (5) see [6, 8, 9]. Hemite-Hdmd inequlities fo log-convex functions wee poved by Dgomi nd Mond [3]. Pchptte [6, 7] lso gve some othe efinements of these inequlities elted to diffeentible log-convex functions. Noo [5] povedthe following Hemite-Hdmd inequlities fo the peinvex nd log-peinvex functions, espectively. (i) Let f : K = [, + η(b, )] (, ) be peinvex function on the intevl of el numbes K (inteio of K) nd, b K with < + η(b, ). Then,the following inequlities hold: f( + η (b, ) ) b η (b, ) f () +f f (x) dx. (6) (ii) Let f be log-peinvex function defined on the intevl [,+η(b,)].then, b η (b, ) f () f f (x) dx log f () log f =L(f(),f), whee L(p, q)(p =q)is the logithmic men of positive el numbes p, q. (7) 3. Min Results Theoem 4. Let f : K = [, + η(b, )] (, ) be n -peinvexfunctionontheintevlofelnumbesk (inteio of K) nd, b K with < + η(b, ). Then,thefollowing inequlities hold: +η(b,) η (b, ) f (x) dx [ f () +f / ],. (8) Poof. Fom Jensen s inequlity, we obtin ( +η(b,) η (b, ) f (x) dx) +η(b,) η (b, ) f (x) dx. (9) Since f is peinvex, then, using Hemite-Hdmd inequlity fo peinvex functions (see [5]), we hve Hence, +η(b,) η (b, ) +η(b,) η (b, ) This completes the poof. f (x) dx f () +f. () f (x) dx[ f () +f / ]. () Coolly 5. Let f : K = [, + η(b, )] (, ) be -peinvexfunctionontheintevlofelnumbesk (inteio of K)nd, b K with < + η(b, ).Then, +η(b,) η (b, ) f (x) dx f () +f. () Theoem 6. Let f : K = [, + η(b,)] (, ) be n -peinvex function (with ) on the intevl of el numbes K (inteio of K)nd, b K with < + η(b, ). Then, the following inequlities hold: +η(b,) η (b, ) f (x) dx [f+ () f + { + f () f ], = { f () f, =. { log f () log f (3) Poof. Fo =,Noo[5] poved this esult. We poceed fo the cse >.Sincef is -peinvex function, fo ll t [,], we hve f(+tη(b, )) (( t) f () +tf ) /. (4)

3 Jounl of Applied Mthemtics 3 Theefoe, +η(b,) η (b, ) f (x) dx = f(+tη(b, ))dt (f () +t(f f ())) / dt. Putting u=f () + t(f f ()),wehve +η(b,) f η (b, ) f (x) dx f f () (u) / dt which completes the poof. = + f () f + f + () f f, () (5) (6) Note tht fo =,intheoem 6, wehvethesme inequlity gin s in Coolly 5. Theoem 7. Let f : K = [, + η(b,)] (, ) be n -peinvex function with s on the intevl of el numbes K (inteio of K)nd, b K with < + η(b, ). Then, f is s-peinvex function. Poof. To pove this, we need the following inequlity: u t V t (( t) u +tv ) / (( t) u s +tv s ) /s, (7) (see []) whee t, s.sincef is -peinvex, by inequlity (7), fo ll u, V K, t [,],weobtin f(u+tη(v,u)) (( t) f (u) +tf (V)) / { (( t) f s (u) +tf s (V)) /s, (f (u)) t (f (V)) t { { (( t) f s (u) +tf s (V)) /s, Hence, f is s-peinvex function. < s = s. (8) As specil cse of Theoem 7, wededucethefollowing esult. Coolly 8. Let f:k=[, + η(b, )] (, ) be n -peinvex function with on the intevl of el numbes K (inteio of K)nd, b K with < + η(b, ). Then, f is peinvex function. Theoem 9. Let f : K = [, + η(b,)] (, ) be n -peinvex function with s on the intevl of el numbes K (inteio of K)nd, b K with < + η(b, ). Then, the following inequlities hold: +η(b,) η (b, ) f (x) dx [f+ () f + + f () f ] s [fs+ () f s+ { s+ f s () f s ], <s f () f log f () log f { s [fs+ () f s+ { s+ f s () f s ], =s. (9) Poof. The left side of these inequlities is cle fom Theoem 7.Fo the ight hnd side,we hve fom inequlity (7) (( t) f () +tf ) / (( t) f s () +tf s ) /s. () Integting both sides with espect to t,weobtin f + () f + + f () f s [fs+ () f s+ s+ f s () f s ]. () Agin, using the othe pt of (7) nd integting, one cn esily deduce the second pt of the equied inequlities. Coolly. Let f : K = [, + η(b, )] (, ) be n -peinvex function with on the intevl of el numbes K (inteio of K)nd, b K with < + η(b, ). Then, the following inequlities hold: f( + η (b, ) ) +η(b,) η (b, ) f () f (x) dx[f() f] ln f [f+ () f + + f () f ] f () +f. () Theoem. Let f,g : K = [, + η(b,)] (, ) be -peinvex nd s-peinvex functions, espectively, with, s >

4 4 Jounl of Applied Mthemtics on the intevl of el numbes K (inteio of K)nd, b K with < + η(b, ).Then,thefollowinginequlitieshold: +η(b,) η (b, ) f (x) g (x) dx [f+ () f + + f () f ] + s [gs+ () g s+ s+ g s () g s ] (f () =f, g() =g). (3) Poof. Since f is -peinvex nd g is s-peinvex, fo ll t [, ],wehve Theefoe, f(+tη(b, )) (( t) f () +tf ) /, g(+tη(b, )) (( t) g s () +tg s ) /s. +η(b,) η (b, ) f (x) g (x) dx = f(+tη(b, ))g(+tη(b, ))dt [( t) f () +tf ] / [( t) g s () +tg s ] /s dt = [f () +t(f f ())] / [g s () +t(g s g s ())] /s dt. Now, pplying Cuchy s inequlity, we obtin [f () +t(f f ())] / [g s () +t(g s g s ())] /s dt [f () +t(f f ())] / dt + [g s () +t(g s g s ())] /s dt = [f+ () f + + f () f ] + s [gs+ () g s+ s+ g s () g s ], which leds us to the equied esult. (4) (5) (6) Note. By putting f=g, =s=,intheoem,wehvethe following inequlity: +η(b,) η (b, ) f (x) dx f () +f. (7) Theoem,fo=s=,wspovedbyNoo[3]. Theoem. Let f,g : K = [, + η(b,)] (, ) be -peinvex ( >)nd-peinvex functions, espectively, on the intevl of el numbes K (inteio of K) nd, b K with < + η(b, ).Then,thefollowinginequlitieshold: +η(b,) η (b, ) f (x) g (x) dx [f+ () f + + f () f ] (8) + 4 [ g () g log g () log g ] (f () =f). Poof. Since f is -peinvex nd g is -peinvex, fo ll t [, ],wehve f(+tη(b, )) (( t) f () +tf ) /, g(+tη(b, )) (g()) t (g ) t. Theefoe, +η(b,) η (b, ) f (x) g (x) dx = f(+tη(b, ))g(+tη(b, ))dt [( t) f () +tf ] / [(g()) t (g ) t ]dt = [f () +t(f f ())] / [(g()) t (g ) t ]dt. Now, pplying Cuchy s inequlity, we obtin [f () +t(f f ())] / [(g ()) t (g ) t ]dt [f () +t(f f ())] / dt + [(g ()) t (g ) t ]dt (9) (3) = [f+ () f + + f () f ]+ 4 [ g () g log (g () /g ) ], (3) which is the equied esult.

5 Jounl of Applied Mthemtics 5 Acknowledgment The uthos would like to thnk efeees of this ppe fo thei insightful comments which getly impoved the entie pesenttion of the ppe. They would lso like to cknowledge Pof. D. Ehsn Ali, VC AWKUM, fo the finncil suppot on the publiction of this ticle. Refeences [] M. A. Hnson, On sufficiency of the Kuhn-Tucke conditions, Jounl of Mthemticl Anlysis nd Applictions, vol.8,no., pp , 98. []M.A.Noo, Vitionl-likeinequlities, Optimiztion, vol. 3, no. 4, pp , 994. [3] M. A. Noo, Invex equilibium poblems, Jounl of Mthemticl Anlysis nd Applictions, vol.3,no.,pp , 5. [4] T. Wei nd B. Mond, Pe-invex functions in multiple objective optimiztion, Jounl of Mthemticl Anlysis nd Applictions, vol. 36, no., pp. 9 38, 988. [5] M. A. Noo, Hemite-Hdmd integl inequlities fo logpeinvex functions, Jounl of Mthemticl Anlysis nd Appoximtion Theoy,vol.,no.,pp.6 3,7. [6] B. G. Pchptte, Mthemticl Inequlities, vol. 67ofNoth- Hollnd Mthemticl Liby, Elsevie, Amstedm, The Nethelnds, 5. [7] C. E. M. Pece, J. Pečić, nd V. Šimić, Stolsky mens nd Hdmd s inequlity, Jounl of Mthemticl Anlysis nd Applictions, vol., no., pp. 99 9, 998. [8] N.P.N.Ngoc,N.V.Vinh,ndP.T.T.Hien, Integlinequlities of Hdmd type fo -convex functions, Intentionl Mthemticl Foum. Jounl fo Theoy nd Applictions,vol.4,no ,pp.73 78,9. [9] G.-S. Yng nd D.-Y. Hwng, Refinements of Hdmd s inequlity fo -convex functions, Indin Jounl of Pue nd Applied Mthemtics,vol.3,no.,pp ,. []G.Zbndn,A.Bodghi,ndA.Kılıçmn, The Hemite- Hdmd inequlity fo -convex functions, Jounl of Inequlities nd Applictions,vol.,ticle5,. [] M. Bessenyei, Hemite-Hdmd-type inequlities fo genelized 3-convex functions, Publictiones Mthemtice Debecen,vol.65,no.-,pp.3 3,4. [] T. Antczk, Men vlue in invexity nlysis, Nonline Anlysis. Theoy, Methods & Applictions,vol.6,no.8,pp , 5. [3] X. M. Yng, X. Q. Yng, nd K. L. Teo, Genelized invexity nd genelized invint monotonicity, Jounl of Optimiztion Theoy nd Applictions,vol.7,no.3,pp.67 65,3. [4] S. R. Mohn nd S. K. Neogy, On invex sets nd peinvex functions, Jounl of Mthemticl Anlysis nd Applictions, vol. 89, no. 3, pp. 9 98, 995. [5] M.A.NoondK.I.Noo, Somechcteiztionsofstongly peinvex functions, Jounl of Mthemticl Anlysis nd Applictions,vol.36,no.,pp ,6. [6] M. A. Noo nd K. I. Noo, Hemiequilibium-like poblems, Nonline Anlysis. Theoy, Methods & Applictions,vol.64,no., pp , 6. [7] B. G. Pchptte, A note on integl inequlities involving two log-convex functions, Mthemticl Inequlities & Applictions,vol.7,no.4,pp.5 55,4. [8] J. Chudzik nd J. Tbo, Chcteiztion of condition elted to clss of peinvex functions, Nonline Anlysis. Theoy, Methods & Applictions, vol.74,no.6,pp ,. [9] C.-P. Liu, Some chcteiztions nd pplictions on stongly α-peinvex nd stongly α-invex functions, Jounl of Industil nd Mngement Optimiztion,vol.4,no.4,pp ,8. [] M. Soleimni-Dmneh, Nonsmooth optimiztion using Modukhovich s subdiffeentil, SIAM Jounl on Contol nd Optimiztion,vol.48,no.5,pp ,. [] M. Soleimni-Dmneh, Genelized invexity in sepble Hilbet spces, Topology,vol.48,no.-4,pp.66 79,9. [] M. Soleimni-Dmneh, The gp function fo optimiztion poblems in Bnch spces, Nonline Anlysis. Theoy, Methods & Applictions,vol.69,no.,pp.76 73,8. [3] M. Soleimni-Dmneh nd M. E. Sbi, Sufficient conditions fo nonsmooth -invexity, NumeiclFunctionlAnlysisnd Optimiztion, vol. 9, no. 5-6, pp , 8. [4] M. Soleimni-Dmneh, Optimlity fo nonsmooth fctionl multiple objective pogmming, Nonline Anlysis. Theoy, Methods & Applictions,vol.68,no.,pp ,8. [5] J. Yng nd X. Yng, Two new chcteiztions of peinvex functions, Dynmics of Continuous, Discete & Impulsive Systems B,vol.9,no.3,pp.45 4,. [6] T. Antczk, -peinvexity nd -invexity in mthemticl pogmming, Computes & Mthemtics with Applictions, vol. 5, no. 3-4, pp , 5. [7] T. Antczk, A new method of solving nonline mthemticl pogmming poblems involving -invex functions, Jounl of Mthemticl Anlysis nd Applictions,vol.3,no.,pp.33 33, 5. [8] S. S. Dgomi nd C. E. M. Pece, Selected Topics on Hemite- Hdmd Type Inequlities, RGMIA Monogph, Victoi Univesity, Melboune, Austli,. [9] J. E. Pečić, F. Poschn, nd Y. L. Tong, Convex Functions, Ptil Odeings nd Sttisticl Applictions, vol.87ofmthemtics in Science nd Engineeing, Acdemic Pess, New Yok, NY, USA, 99. [3] S. S. Dgomi nd B. Mond, Integl inequlities of Hdmd type fo log-convex functions, Demonsttio Mthemtic, vol. 3, no., pp , 998. [3] M. A. Noo, On Hdmd integl inequlities involving two log-peinvex functions, Jounl of Inequlities in Pue nd Applied Mthemtics,vol.8,no.3,ticle75,7.

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About Some Inequalities for Isotonic Linear Functionals and Applications

About Some Inequalities for Isotonic Linear Functionals and Applications Applied Mthemticl Sciences Vol. 8 04 no. 79 8909-899 HIKARI Ltd www.m-hiki.com http://dx.doi.og/0.988/ms.04.40858 Aout Some Inequlities fo Isotonic Line Functionls nd Applictions Loedn Ciudiu Deptment