NEW INTEGRAL INEQUALITIES OF THE TYPE OF SIMPSON S AND HERMITE-HADAMARD S FOR TWICE DIFFERENTIABLE QUASI-GEOMETRICALLY CONVEX MAPPINGS

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1 TJMM 8 6, No., NEW INTEGRAL INEQUALITIES OF THE TYPE OF SIMPSON S AND HERMITE-HADAMARD S FOR TWICE DIFFERENTIABLE QUASI-GEOMETRICALLY CONVEX MAPPINGS MUHAMMAD MUDDASSAR AND ZAFFER ELAHI Astrct. In this pper, we define new identity for twice differentie ppings nd otined soe new estites on the generiztion of Hdrd s nd Sipson s type ineuities for usi-geoetricy convex ppings using of this identity.. Introction nd Preiinry Resuts A convex function is continuous function whose vue t the idpoint of every interv in its doin does not exceed the rithetic en of its vue t the ends of the interv. An iportnt thetic proe is to investigte how function ehves under the ction of ens. The est known cse is tht of idpoint convex or Jensen convex functions, which des with the rithetic en [7, pp.. More genery, function fx is convex on n interv [,, if for ny two points x, y [, nd α, β,, we hve fαx βy αfx βfy A re vued function defined on nonepty suinterv I of R is ced convex if we repce α β for points x nd y I in ove ineuity. It is ced stricty convex if the ove ineuity hods stricty whenever x nd y re distinct points. If f is convex respectivey, stricty convex then we sy tht f is concve respectivey, stricty concve. A function is ced ffine if it is oth convex nd concve. The ppernce of the new thetic ineuity often puts on fir founion for the heuristic goriths nd proceres used in ppied sciences. Aong others one of the in ineuity, which gives us n expicit error ounds in the trpezoid nd idpoint rues of sooth function, ced Herit-Hdrd s ineuity defined s f f f ft f : [, R is convex function. Both ineuities hod in the reversed direction for f to e concve. We note tht Herit-Hdrd s ineuity y e regrded s refineent of the concept of convexity nd it foows esiy fro Jensens ineuity. Ineuity hs received renewed ttention in recent yers nd rerke vriety of refineents nd generiztions hve een found in iterture nd the references cited therein. Mthetics Suject Cssifiction. 6D, 6D5, 39A. Key words nd phrses. usi-geoetricy convex functions, herite hdrd type ineuities, sipson type ineuity. 37

2 38 M. MUDDASSAR AND Z. ELAHI The second we known ineuity in iterture s Sipson s Ineuity defined s [ f ff fxdx 3 88 f iv 4 f : [, R is four ties continuous differentie pping on, nd f iv sup x, f iv x <. It is we known tht if the pping f is neither four ties differentie nor is the fourth derivtive f iv ounded on,, then we cnnot ppy the cssic Sipson udrture foru. The notion of usi-convex functions generizes the notion of convex functions. More excty, function f : [, R is sid usi-convex on [, if fαx βy supfx, fy x, y [,, α, β, nd α β. The notion of geoetricy convex first introce y Nicuescu, C. P. in [5 nd [6 nd proced s Definition. A function f : I R R is sid to e GG-convex ced geoetricy convex function if f x α y β f α xf β y x, y [,, α, β, nd α β. Nicuescu in se rtice defined the ter geoetric rithticy convex with nottion GA-convex s Definition. A function f : I R R is sid to e GG-convex ced geoetricy convex function if f x α y β αfx βfy x, y [,, α, β, nd α β. In [, İşcn, İ. gve definition of usi-geoetricy convexity s foows: Definition 3. A function f : I R R is sid to e usi-convex if f x α y β supfx, fy x, y [,, α, β, nd α β. Cery, ny GA-convex nd geoetricy convex functions re usi-geoetricy convex functions. Furtherore, there exist usi-geoetricy convex functions which re neither GA-convex nor GG-convex [. Recenty, İşcn, İ. et. in [4 estished soe resuts sed on singe differentiiity for usi-geoetricy convex functions using the identity Le. A function f : I R R e differentie function on I o such tht f L [,,, I with <. Then for λ, R, we hve: I f λ,,, t t t f t t t λ t t f t t I f λ,,, λ f f λ f, I with < nd λ, R. fu u 3

3 NEW INTEGRAL INEQUALITIES OF THE TYPE OF SIMPSON S AND This rtice is in the continution of [4. The in purpose of this rtice is to estish soe new gener integr ineuities of Herite-Hdrd nd Sipson type for twice differentie usi-geoetricy convex functions y using new integr identity.. Min Resuts In order to prove our in resuts we need the foowing identity. Le. A function f : I R R e differentie function on I o such tht f L [,,, I with <. Then for λ, R, we hve M f λ,,, tt t t f t t tt λ t t f t t M f λ,,, λ f f λ f λ f /, I with < nd λ, R. fu u Proof. Using integrtion rues nd chnging the preter, we cn esiy prove the ove resut. Theore. A function f : I R R e twice differentie function on I o such tht f L [,,, I with <. If f is usi geoetricy convex on [, for soe fixed nd / λ, then the foowing ineuity hods M f λ,,, c sup f, f c c 3,,, c λc 4 λ,,, c λ λ3 3 λ 7λ 8 [ 4 c 3,,, 8 L, 8 L, L, 4 5

4 4 M. MUDDASSAR AND Z. ELAHI c 4 λ,,, [ λ λ 4λ λ λ L λ, λ λ L, 8 λ λ λ 4 4 Proof. Since f is usi-geoetricy convex on [, for t [. f t t sup f, f Fro e nd using power en ineuity, we hve M f λ,,, tt Let here sup f, f tt λ c c λ c tt tt λ tt t t sup f, f λ t t tλ t λ t t tt λ tt tt λ λ3 3 λ 7λ 8 4 3,,, t t t t t t t t tt t t Using sustitution u t t in c 3,,,, we hve t t t t 3 u u t t t t 3 3 tt t t u u u 3 u u 3 3 u

5 NEW INTEGRAL INEQUALITIES OF THE TYPE OF SIMPSON S AND... 4 finy, we get And tt t t c 3,,, c 4 λ,,, 8 3 u [ 8 L, 8 L, λ L, tt λ t t t t t t λ tt t t Using se sustitution u t t in c 4 [ λ c 4 λ,,, This copetes the proof. λ,,,, we hve λ 4λ λ λ L λ, λ λ L, 8 λ λ λ 4 4 Corory. A function f : I R R e twice differentie function on I o such tht f L [,,, I with <. If f is usi geoetricy convex on [, for soe fixed nd for, R with <, then the foowing ineuity hods f,, sup f, f,,, c 4,,, c 3 f,, f f f fu u 7

6 4 M. MUDDASSAR AND Z. ELAHI nd c c 3,,, c 4,,, c [ 3 8 L, L, [ 8 L, L, Proof. Proof is exchngee with Theore with the sustitution nd λ. Theore. A function f : I R R e twice differentie function on I o such tht f L [,,, I with <. If f is usi geoetricy convex on [, for soe fixed conjugte nuers p, > nd / λ, then the foowing ineuity hods M f λ,,, sup f, f c p 5 p, c 7,, c p 6 p, λc 8,, 8 c 5 p, [ p 4 p p λ p, c 6 p, λ λ4 p p c 7,, L,, c 8,, L, Proof. Fro e y ppying Höder ineuity nd using the usi geoetricy convexity on [, of f, we hve M f λ,,, t t p p sup f, f tt λ p t t p t t sup f, f 9

7 NEW INTEGRAL INEQUALITIES OF THE TYPE OF SIMPSON S AND M f λ,,, sup f, f t t p t t p tt λ p t t c 5 p, c 6 p, t t p tλ t p λ t p t p t p t λ p Here we use u t t to ccute c 7,, nd c 8,, c 7,, t t L, c 8,, Hence 8 esiy found fro. p t p t p [ p 4 p p t t L, λ t p t λ p λ p λ4 p p Corory. A function f : I R R e twice differentie function on I o such tht f L [,,, I with <. If f is usi geoetricy convex on [, for soe fixed conjugte nuers p, with > nd for, R with <, then the foowing ineuity hods [ p f,, c 5 p, p p c 6 p, 4 p sup f, f c 7,, c 8,, p p [ p 4 p c 7,, L,, c 8,, L, nd f,, fixed in Corory. Proof. Proof is exchngee with Theore with the sustitution nd λ. Theore 3. A function f : I R R e twice differentie function on I o such tht f L [,,, I with <. If f is usi geoetricy convex on [, for soe fixed conjugte nuers p, > nd / λ, then the foowing ineuity hods M f λ,,, sup f, f, c p 7 p,, c 5, c p 8 p,, c 6, λ

8 44 M. MUDDASSAR AND Z. ELAHI nd p c 5, [ 4 λ, c 6, λ λ4 c 7 p,, p L p, p, c 8 p,, p L p, p Proof. Fro e y ppying Höder ineuity nd using the usi geoetricy convexity on [, of f, we hve M f λ,,, t t p p t t sup f, f t t p p tt λ sup f, f M f λ,,, sup f, f t t t t p p tt λ t t p p 3 4 c 5, c 6 p, λ t t [ 4 tλ t λ λ4 Here we use u t t to ccute c 7,, nd c 8,, c 7 p,, t t p p L p, p c 8 p,, Hence esiy found fro 4. t t p p L p, p Corory 3. A function f : I R R e twice differentie function on I o such tht f L [,,, I with <. If f is usi geoetricy convex on [, for soe fixed conjugte nuers p, nd for, R with <, then the foowing ineuity hods f,, [ 4 sup f, f c p 7 p,, c p 8 p,, 5

9 NEW INTEGRAL INEQUALITIES OF THE TYPE OF SIMPSON S AND c 5, c 6, [ 4 c 7 p,, p L p, p, c 8 p,, p L p, p nd f,, fixed in Corory. Proof. Proof is exchngee with Theore 3 with the sustitution nd λ. 3. Copeting Interests The uthors decre tht they hve no copeting interests. The uthors so decre tht they hve no finnci nd non-finnci copeting interests. 4. Acknowedgeents This reserch pper is de possie through the hep nd support fro University of Engineering & Technoogy, Txi. We grtefuy cknowedge the tie nd expertise devoted to reviewing ppers y the dvisory editors, the eers of the editori ord, nd the referees. Finy, we woud ike to thnk Mr. Mhood Akhter Chirn, Deprtent of Bsic Sciences & Hunities, UET, Txi for his ost support nd encourgeent. References [ İşcn, İ., New gener integr ineuities for usi-geoetricy convex functions vi frction integrs, Journ of Ineuities nd Appictions 49, 3. [ İşcn, İ., Soe New Herite-Hdrd Type Ineuities for Geoetricy Convex Functions, Mthetics nd Sttistics, 86 9, 3. [3 İşcn, İ., Soe Generized Herite-Hdrd Type Ineuities for Qusi-Geoetricy Convex Functions, Aericn Journ of Mthetic Anysis 3, 48 5, 3. [4 İşcn, İ., Keri Bekr nd Sei Nun, Herite-Hdrd nd Sipson Type Ineuities for Differentie Qusi-Geoetricy Convex Functions, Turkish Journ of Anysis nd Nuer Theory, 4, Vo., No., [5 Nicuescu, C.P., Convexity ccording to the geoetric en, Mth. Ineu. App. 3, [6 Nicuescu, C.P., Convexity ccording to ens, Mth. Ineu. App [7 Nicuescu, C.P nd Persson, L.E., Convex functions nd their ppictions. A Conteporry Approch, CMS Books in Mthetics vo.3, Springer-Verg, New York, 6. Deprtent of Mthetics, University of Engineering nd Technoogy, Txi, Pkistn E-i ddress: ik.uddssr@gi.co Deprtent of Mthetics, University of Engineering nd Technoogy, Txi, Pkistn E-i ddress: zfferehi@hoti.co,