On Hermite-Hadamard type integral inequalities for functions whose second derivative are nonconvex

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1 Mly J Mt On Hermite-Hdmrd tye integrl ineulities for functions whose second derivtive re nonconvex Mehmet Zeki SARIKAYA, Hkn Bozkurt nd Mehmet Eyü KİRİŞ b Dertment of Mthemtics, Fculty of Science nd Arts, Düzce University, Düzce-TURKEY b Dertment of Mthemtics, Fculty of Science nd Arts, Afyon Koctee University, Afyon-TURKEY Abstrct In this er, we extend some estimtes of the right hnd side of Hermite- Hdmrd tye ineulity for nonconvex functions whose second derivtives bsolute vlues re ϕ-convex, log-ϕ-convex, nd usi-ϕconvex Keywords: ineulity Hermite-Hdmrd s ineulities, ϕ-convex functions, log-ϕ-convex, usi-ϕ-convex, Hölder s MSC: 6D7, 6D, 6D99 c MJM All rights reserved Introduction It is well known tht if f is convex function on the intervl I =, b] nd, b I with < b, then + b f b f x dx b f + f b which is known s the Hermite-Hdmrd ineulity for the convex functions Both ineulities hold in the reversed direction if f is concve We note tht Hermite-Hdmrd ineulity my be regrded s refinement of the concet of convexity nd it follows esily from Jensen s ineulity Hermite-Hdmrd ineulity for convex functions hs received renewed ttention in recent yers nd remrkble vriety of refinements nd generliztions hve been found see, for exmle, ]-4], ]-] The following lemm ws roved for twice differentible mings in 3]: Lemm Let f : I R R be twice differentible ming on I o,, b I with < b nd f of integrble on, b], the following eulity holds: f + f b + b b f x dx = b t t f t + t b dt A simle roof of this eulity cn be lso done by twice integrting by rts in the right hnd side In 4], by using Lemm, Hussin et l roved some ineulities relted to Hermite-Hdmrd s ineulity for s-convex functions: Corresonding uthor E-mil ddress: srikymz@gmilcom Mehmet Zeki SARIKAYA

2 94 Mehmet Zeki SARIKAYA et l / The Hermite-Hdmrd s ineulities Theorem Let f : I, R be twice differentible ming on I such tht f L, b], where, b I with < b If f is s convex on, b] for some fixed s, ] nd, then the following ineulity holds: where + = f + f b Remrk If we tke s = in, then we hve f + f b b b f xdx b 6 b b f xdx b f + f b ], s + s + 3 f + f b ] We recll tht the notion of usi-convex functions generlizes the notion of convex functions More recisely, function f :, b] R R is sid usi-convex on, b] if f tx + ty su { f x, f y} for ll x, y, b] nd t, ] Clerly, ny convex function is usi-convex function Furthermore, there exist usi-convex functions which re not convex see ] Alomri, Drus nd Drgomir in ] introduced the following theorems for twice differentible usiconvex functions: Theorem Let f : I R R be twice differentible function on I o,, b I o with < b nd f is integrble on, b] If f is usiconvex on, b], then the following ineulity holds f + f b b b f x dx b mx { f, f b } Theorem 3 Let f : I R R be twice differentible function on I o,, b I o with < b nd f is integrble on, b] If f is usiconvex on, b], for >, then the following ineulity holds f + f b b f x dx b where + = b Γ + Γ 3 + mx { f, f b }, Theorem 4 Let f : I R R be twice differentible function on I o,, b I o with < b nd f is integrble on, b] If f is usiconvex on, b], for, then the following ineulity holds f + f b b b f x dx b mx { f, f b } Preliminries Let f, ϕ : K R, where K is nonemty closed set in R n, be continuous functions First of ll, we recll the following well known results nd concets, which re minly due to Noor nd Noor 5] nd Noor 9] s follows: Definition Let u, v K Then the set K is sid to be ϕ convex t u with resect to ϕ, if u + te iϕ v u K, u, v K, t, ]

3 Mehmet Zeki SARIKAYA et l / The Hermite-Hdmrd s ineulities 95 Remrk We would like to mention tht Definition of ϕ-convex set hs cler geometric interrettion This definition essentilly sys tht there is th strting from oint u which is contined in K We do not reuire tht the oint v should be one of the end oints of the th This observtion lys n imortnt role in our nlysis Note tht, if we demnd tht v should be n end oint of the th for every ir of oints, u, v K, then e iϕ v u = v u if nd only if, ϕ =, nd conseuently ϕ-convexity reduces to convexity Thus, it is true tht every convex set is lso n ϕ-convex set, but the converse is not necessrily true, see 5]-9] nd the references therein Definition The function f on the ϕ-convex set K is sid to be ϕ-convex with resect to ϕ, if f u + te iϕ v u t f u + t f v, u, v K, t, ] The function f is sid to be ϕ-concve if nd only if f is ϕ-convex Note tht every convex function is ϕ-convex function, but the converse is not true Definition 3 The function f on the ϕ-convex set K is sid to be logrithmic ϕ-convex with resect to ϕ, such tht f u + te iϕ v u f u t f v t, u, v K, t, ] where f > Now we define new definition for usi-ϕ-convex functions s follows: Definition 4 The function f on the usi ϕ-convex set K is sid to be usi ϕ-convex with resect to ϕ, if f u + te iϕ v u mx { f u, f v} From the bove definitions, we hve f u + te iϕ v u f u t f v t t f u + t f v mx { f u, f v} Clerly, ny ϕ-convex function is usi ϕ-convex function Furthermore, there exist usi ϕ-convex functions which re neither ϕ-convex nor continuous For exmle, for { k, uv, k Z ϕv, u = k, uv <, k Z the floor function f loor x = x, is the lrgest integer not greter thn x, is n exmle of monotonic incresing function which is usi ϕ-convex but it is neither ϕ-convex nor continuous In 7], Noor roved the Hermite-Hdmrd ineulity for the ϕ convex functions s follows: Theorem 5 Let f : K =, + e iϕ b ], be ϕ-convex function on the intervl of rel numbers K the interior of K nd, b K with < + e iϕ b nd ϕ Then the following ineulity holds: f + e iϕ b e iϕ b +e iϕ b f + f + e iϕ b f x dx 3 f + f b This ineulity cn esily show tht using the ϕ-convex function s definition nd f + e iϕ b < f b In 9] nd ], the uthors roved some generliztion ineulities connected with Hermite-Hdmrd s ineulity for diferentible ϕ-convex functions In this rticle, using functions whose second derivtives bsolute vlues re ϕ-convex, log-ϕ-convex nd usi-ϕ-convex, we obtined new ineulities relted to the right side of Hermite-Hdmrd ineulity given with 3

4 96 Mehmet Zeki SARIKAYA et l / The Hermite-Hdmrd s ineulities 3 Hermite-Hdmrd Tye Ineulities We will strt the following theorem: Theorem 36 Let K R be n oen intervl,, + e iϕ b K with < b nd f : K =, + e iϕ b ], twice differentible ming such tht f is integrble nd ϕ If f is ϕ-convex function on, + e iϕ b ] Then, the following ineulity holds: e iϕ b eiϕ b 4 +e iϕ b f + f b ] f xdx f + f + eiϕ b Proof If the rtil integrtion method is lied twice, then it follows tht e iϕ b t t f + te iϕ b dt 34 = +e iϕ b e iϕ b f xdx f + f + eiϕ b Thus, by ϕ-convexity function of f, we hve e iϕ b eiϕ b eiϕ b eiϕ b 4 +e iϕ b f xdx f + f + eiϕ b t t f + te iϕ b dt t t t f + t f b ] dt f + f b ] which the roof is comleted Theorem 37 Let f : K =, + e iϕ b ], be twice differentible ming on K nd f be integrble on, + e iϕ b ] Assume R with > If f / is ϕ-convex function on the intervl of rel numbers K the interior of K nd, b K with < + e iϕ b nd ϕ Then, the following ineulity holds: e iϕ b eiϕ b +e iϕ b f xdx f + f + eiϕ b Γ + f + f b Γ 3 +

5 Mehmet Zeki SARIKAYA et l / The Hermite-Hdmrd s ineulities 97 Proof By ssumtion, Hölder s ineulity nd 34, we hve e iϕ b eiϕ b eiϕ b eiϕ b = eiϕ b +e iϕ b f xdx f + f + eiϕ b t t f + te iϕ b dt t t dt f + te iϕ b Γ + Γ 3 + Γ + f + f b Γ 3 + dt t f + t f b ] dt where we use the fct tht t t dt = Γ + Γ 3 + which comletes the roof Let us denote by A, b the rithmetic men of the nonnegtive rel numbers, nd by L, b the logritmic men of the sme numbers Theorem 3 Let K R be n oen intervl,, + e iϕ b K with < b nd f : K =, + e iϕ b ], twice differentible ming such tht f is integrble nd ϕ If f is log ϕ-convex function on, + e iϕ b ] Then, the following ineulity holds: +e iϕ b e iϕ f xdx f + f + eiϕ b b e iϕ b A f b, f L f b, f ] log f b log f Proof By using 34 nd log ϕ-convexity of f, we hve e iϕ b eiϕ b +e iϕ b f xdx f + f + teiϕ b t t f + te iϕ b dt eiϕ b t t f t f b t dt ] = eiϕ b f b + f log f b log f f b f log f b log f 3 e iϕ b A = f log f b log f b, f L f b, f ] The roof of Theorem 3 is comleted Theorem 39 Let f : K =, + e iϕ b ], be twice differentible ming on K o nd f be integrble on, + e iϕ b ] Assume R with > If f / is log ϕ-convex function on the intervl of rel numbers

6 9 Mehmet Zeki SARIKAYA et l / The Hermite-Hdmrd s ineulities K o the interior of K nd, b K o with < + e iϕ b nd ϕ Then, the following ineulity holds: e iϕ b eiϕ b +e iϕ b f xdx f + f + eiϕ b Γ + Γ 3 + f f b log f b log f Proof By using 34 nd the well known Hölder s integrl ineulity, we obtin e iϕ b eiϕ b eiϕ b eiϕ b = eiϕ b +e iϕ b f xdx f + f + eiϕ b t t f + te iϕ b dt t t dt f + te iϕ b Γ + Γ 3 + Γ + Γ 3 + dt f t f b t dt f f b log f b log f Theorem 3 Let f : K =, + e iϕ b ], be differentible ming on K nd f be integrble on, + e iϕ b ] If f is usi ϕ-convex function on the intervl of rel numbers K o the interior of K nd, b K o with < + e iϕ b nd ϕ Then, the following ineulity holds: e iϕ b eiϕ b 4 +e iϕ b mx{ f, f b } Proof By using 34 nd the usi ϕ-convexity of f, we hve e iϕ b eiϕ b eiϕ b eiϕ b 4 +e iϕ b f xdx f + f + teiϕ b f xdx f + f + teiϕ b t t f + te iϕ b dt mx{ f, f b } t t dt mx{ f, f b } Theorem 3 Let f : K =, + e iϕ b ], be differentible ming on K o nd f be integrble on, + e iϕ b ] Assume R with > If f / is usi ϕ-convex function on the intervl of rel numbers K o the interior of K nd, b K o with < + e iϕ b nd ϕ Then, the following ineulity

7 Mehmet Zeki SARIKAYA et l / The Hermite-Hdmrd s ineulities 99 holds: e iϕ b eiϕ b f xdx f + f + teiϕ b Γ + Γ 3 + mx{ f, f b ] } +e iϕ b Proof By using 34 nd the well known Hölder s integrl ineulity, we get +e iϕ b e iϕ f xdx f + f + teiϕ b b eiϕ b t t f + te iϕ b dt eiϕ b t t dt f + te iϕ b dt eiϕ b Γ + Γ 3 + eiϕ b Γ + Γ 3 + mx{ f, f b }dt mx{ f, f b ] } Theorem 3 Let f : K =, + e iϕ b ], be differentible ming on K o nd f be integrble on, + e iϕ b ] Assume R with If f is usi ϕ-convex function on the intervl of rel numbers K o the interior of K nd, b K o with < + e iϕ b nd ϕ Then, the following ineulity holds: +e iϕ b e iϕ f xdx f + f + teiϕ b b eiϕ b mx{ f, f b }] Proof By using nd the well known ower men integrl ineulity, we hve +e iϕ b e iϕ f xdx f + f + teiϕ b b eiϕ b t t f + te iϕ b dt eiϕ b t t dt t t f + te iϕ b dt where + = eiϕ b eiϕ b mx{ f, f b } 6 mx{ f, f b }], t t dt References ] M Alomri, M Drus nd S S Drgomir, New ineulities of Hermite-Hdmrd s tye for functions whose second derivtives bsolute vlues re usiconvex, Tmk J Mth, 4,

8 3 Mehmet Zeki SARIKAYA et l / The Hermite-Hdmrd s ineulities ] S S Drgomir nd R P Agrwl, Two ineulities for differentible mings nd lictions to secil mens of rel numbers nd trezoidl formul, Al Mth Lett, 599, ] S S Drgomir nd C E M Perce, Selected Toics on Hermite-Hdmrd Ineulities nd Alictions, RGMIA Monogrhs, Victori University, 4] S Hussin, M I Bhtti nd M Ibl, Hdmrd-tye ineulities for s-convex functions I, Punjb Univ Jour of Mth, 49, 5-6 5] M A Noor, Some new clsses of nonconvex functions, Nonl Funct Anl Al, 6, ] M A Noor, On Hdmrd integrl ineulities involving two log-reinvex functions, J Ineul Pure Al Mth, 37, -6 7] M A Noor, Hermite-Hdmrd integrl ineulities for log-ϕ-convex functions, Nonliner Anlysis Forum, 3, 9 4 ] M A Noor, On clss of generl vriotionl ineulities, J Adv Mth Studies,, 3-4 9] K I Noor nd M A Noor, Relxed strongly nonconvex functions, Al Mth E-Notes, 66, ] DA Ion, Some estimtes on the Hermite-Hdmrd ineulity through usi-convex functions, Annls of University of Criov Mth Com Sci Ser, 347, -7 ] C E M Perce nd J Pečrić, Ineulities for differentible mings with liction to secil mens nd udrture formule, Al Mth Lett, 3, 5 55 ] J Pečrić, F Proschn nd Y L Tong, Convex functions, rtil ordering nd sttisticl lictions, Acdemic Press, New York, 99 3] A Sglm, M Z Sriky nd H Yildirim, Some new ineulities of Hermite-Hdmrd s tye, Kyungook Mthemticl Journl, 5, ] M Z Sriky, A Sglm nd H Yıldırım, New ineulities of Hermite-Hdmrd tye for functions whose second derivtives bsolute vlues re convex nd usi-convex, Interntionl Journl of Oen Problems in Comuter Science nd Mthemtics IJOPCM, 53 5] M Z Sriky, A Sglm nd H Yıldırım, On some Hdmrd-tye ineulities for h-convex functions, Journl of Mthemticl Ineulities, 3, ] M Z Sriky, M Avci nd H Kvurmci, On some ineulities of Hermite-Hdmrd tye for convex functions, ICMS Iterntionl Conference on Mthemticl Science AIP Conference Proceedings 39, 5 7] M Z Sriky nd N Aktn, On the generliztion some integrl ineulities nd their lictions Mthemticl nd Comuter Modelling, 549-, 75- ] M Z Sriky, E Set nd M E Ozdemir, On some new ineulities of Hdmrd tye involving h- convex functions, Act Mthemtic Universittis Comenine, Vol LXXIX,, ] M Z Sriky, H Bozkurt nd N Al, On Hdmrd Tye Integrl Ineulities for nonconvex Functions, Mthemticl Sciences And Alictions E-Notes, in ress, rxiv:3v ] M Z Sriky, N Al nd H Bozkurt, On Hermite-Hdmrd Tye Integrl Ineulities for reinvex nd log-reinvex functions, Contemorry Anlysis nd Alied Mthemtics, 3, 37-5 Received: December 3, 3; Acceted: Aril 5, 4 UNIVERSITY PRESS Website: htt://wwwmlyjournlorg/

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