Journl of Mthemticl Ineulities Volume 1, Number 3 18, 655 664 doi:1.7153/jmi-18-1-5 NEW HERMITE HADAMARD INEQUALITIES VIA FRACTIONAL INTEGRALS, WHOSE ABSOLUTE VALUES OF SECOND DERIVATIVES IS P CONVEX SHAHID QAISAR, FAROOQ AHMAD, S. S. DRAGOMIR AND MUHAMMAD IQBAL Communicted by A. Vukelić Abstrct. In this reserch rticle, uthors hve estblished generl integrl identity for Riemnn- Liouville frctionl integrls. Some new results relted to the left-hnd side of Hermite-Hdmrd type integrl ineulities utilizing this integrl identity for the clss of functions whose second derivtives t some power re P-convex re obtined.the presented results hve some closely connection with [M. E. Özdemir, C. Yıldız, A. O. Akdemir, E. Set, On some ineulities for s-convex functions nd pplictions, Jounl of Ineulities nd Applictions, 13:333] 1. Introduction Let f : I R R be convex function on the intervl I of rel numbers nd,b I with < b. Then b f 1 b f fb fxdx. 1 b is known in the literture s the Hdmrd ineulity for convex mpping. Both ineulities hold in the reversed direction if f is concve [1].We tke note tht Hdmrd s ineulity might be viewed s refinement of the ide of convexity nd it tkes fter effortlessly from Jensen s ineulity. Note tht few of the clssicl ineulities for mens cn be gotten from 1 for suitble specific selections of the mpping f. It is notble tht the Hermite-Hdmrd ineulity plys n essentil prt in nonliner nlysis. In the course of the most recent decde, this clssicl ineulity hs been enhnced nd summed up in vrious routes; there hve been extensive number of reserch rticles written on this subject, see, [ 15] nd the references therein. Some importnt definitions nd mthemticl preliminries of frctionl clculus theory which re utilized further s bit of this pper. Let I R be n intervl. The function f : I R is sid to be P-convex or belongs to the clss PI if it is nonnegtive nd, for ll,b I nd t [,1], stisfies the ineulity f t1 tb f fb Mthemtics subject clssifiction 1: 6A15, 6A51, 6D1. Keywords nd phrses: Hermite-Hdmrd type ineulity, convex functions, power-men ineulity, Riemnn-Liouville frctionl integrl. c Ð, Zgreb Pper JMI-1-5 655
656 S. QAISAR, F. AHMAD, S. S. DRAGOMIR AND M. IQBAL Note tht PI contin ll nonnegtive convex functions nd usi convex functions. Since then numerous rticles hve ppered in the literture reflecting further pplictions in this ctegory see [3, 6, 1, 15] DEFINITION 1. Let f L 1 [,b]. The left-side nd right-side Riemnn-Liouville frctionl integrls of order > with re defined by J fx = 1 Γ x x t 1 ft, < x nd Jb x = 1 b t x 1 ft, x < b Γ x respectively, where Γ. is Gmm function nd its definition is Γ = e u u 1 du. It is to be noted tht J fx = Jb x = fx. In the cse of = 1, the frctionl integrl reduces to the clssicl integrl. Properties relting to this opertor nd for useful detils on Hermite-Hdmrd type ineulities connected with frctionl integrl ineulities, reders re directed to [] [7], [8], [1], [13]. In [13] Sriky et l. proved vrint of Hermite-Hdmrd s ineulities for frctionl integrl which follows s: THEOREM 1. Let f : [,b] R be positive function with < b nd f L 1 [,b]. If f is convex function on [,b], then the following ineulities for frctionl integrls hold: with > b f Γ 1 [ J b fbjb ] f fb REMARK 1. For = 1, ineulity 3 reduces to ineulity 1. In [11] Özdemir et l. proved some ineulities relted to Hermite-Hdmrd s ineulities for functions whose second derivtives in bsolute vlue t certin powers re s-convex functions s follows: THEOREM. Let f : I [, R be twice differentible mpping on I where I is the interior of I such tht f L[,b], where,b I with < b. If f is s-convex on [, b], for some fixed s, 1], then the following ineulity holds: b f 1 b b fxdx b 8s1ss3 f s1s f b [ 1s 1 s ] b f f b } 8s1ss3 3 } f b
NEW RESULTS ON H-H INEQUALITY VIA FRACTIONAL INTEGRALS 657 PROPOSITION 1. Under the ssumptions of Theorem for s = 1 by Theorem, then we get, b f 1 b fxdx b b 48 [ f f b ] 4 THEOREM 3. Let f : I [, R be twice differentible mpping on I where I is the interior of I such tht f L[,b], where,b I with < b. If f, p, 1 is s-convex on [,b], for some fixed s,1], then the following ineulity holds: b f 1 b fxdx b b 1 1/p 16 3 s1ss3 f 1 f b 1/ s3 1 f b } 1/ s3 s1ss3 f b PROPOSITION. Under the ssumptions of Theorem 3 for s = 1 by Theorem 3, then we get, b f 1 b fxdx b b 48 f b 3 1/ 1 4 3 f f b 1/ } 1/ 1 3 f b The im of this pper is to provide unified pproch to estblish Hermite-Hdmrd type ineulities for Riemnn-Liouville frctionl integrl using the convexity s well s concvity, for functions whose bsolute vlues of second derivtives re P-convex. we will derive generl integrl identity for convex functions. 5. Min results To obtin our principl results we need the following Lemm: LEMMA 1. Let I R be n open intervl,,b I with < b nd f : [,b] R be twice differentible function such tht f is integrble nd < 1 on,b with < b. If f is convex on [,b], then the following identity for Riemnn- Liouville frctionl integrls holds: Γ 1 b [ J fbj b ] b b = 3 1 4 k=1 I k,
658 S. QAISAR, F. AHMAD, S. S. DRAGOMIR AND M. IQBAL where I 1 = I = I 3 = I 4 = 1 t 1 f t1 t b, 1 t 1 1 t 1 t f tb1 t b, 1 t 1 f tb1 t b, 1 t 1 1 t 1 t f t1 t b. A simple proof of this ineulity cn be done by integrting by prts. The detils re left to the interested reders. Using Lemm 1 the following results cn be obtined. THEOREM 4. Let I R be n open intervl,,b I with < b nd f : [,b] R be twice differentible function such tht f is integrble nd < 1 on,b with < b. If f is P-convex on [,b], then we hve the following ineulity for Riemnn-Liouville frctionl integrls: Γ 1 [ b fbj b ] b where b 3 1 Q 1 Q Q 1 = Q = 1 t 1 = f f b 1 1 t 1 1 t 1 t = 1 1 1 f b } Proof. Since f, is P-convex function, by using Lemm 1, we get Γ 1 [ b fbjb ] b b 1 3 1 t 1 f t1 t b f tb1 t b 1 b 3 1 f tb1 t b 1 t 1 1 t 1 t f t1 t b } 6 }
NEW RESULTS ON H-H INEQUALITY VIA FRACTIONAL INTEGRALS 659 b 3 1 t 1 f t1 t b f tb1 t b 1 b 1 3 1 t 1 1 t 1 t 1 f tb1 t b f t1 t b b 3 1 Q 1 f f b f b } b 3 1 Q f f b f b } b 3 1 Q 1 Q f f b f b } which completes the proof. COROLLARY 1. Under the ssumption of Theorem 4 if we put f = f b = then ineulity tkes the following form Γ 1 [ b fbjb ] b b 1 Q 1 Q f b The corresponding version for powers of the bsolute vlue of the derivtive is incorported in the following theorem. THEOREM 5. Let I R be n open intervl,,b I with < b nd f : [,b] R be twice differentible function such tht f is integrble nd < 1 on,b with < b. If f is P-convex on [,b], 1 then we hve the following ineulity for Riemnn-Liouville frctionl integrls: Γ 1 [ b fbjb ] b [ b 3 1 Q 1 1 1/ Q 1 f b } ] 1/ f b 3 1 Q 1 1/ [ Q f b f b } 1/ ] 7
66 S. QAISAR, F. AHMAD, S. S. DRAGOMIR AND M. IQBAL Proof. Using the well-known power-men integrl ineulity for > 1, we hve Γ 1 [ b fbjb ] b b 3 1 t 1 f t1 t b 1 b 1 1 1/ 3 1 t 1 1 [ 1 t 1 f t1 t b 1/ 1 t 1 f tb1 t b ] 1/ b 1 1 1/ 3 1 t 1 1 t 1 t 1 [ 1 t 1 1 t 1 t f 1 t 1 1 t 1 t f tb1 t b [ b 3 1 Q 1 1 1/ Q 1 f Q 1 f b ] b } 1/ f [ b 3 1 Q 1 1/ Q f Q f b b } ] 1/ f [ b 3 1 Q 1 1 1/ Q 1 f [ b 3 1 Q 1 1/ Q f b b f f b b f f b t1 t b } 1/ } 1/ } 1/ ] } 1/ ] 1/. ] 1/ Which completes the proof.
NEW RESULTS ON H-H INEQUALITY VIA FRACTIONAL INTEGRALS 661 COROLLARY. On letting = 1 in Theorem 5, then ineulity 7 becomes s: b f 1 b fxdx b b f f b 1/ f b } 1/ f b 48 In the following, we obtin estimte of Hermite-Hdmrd ineulity 3 for concve functions. THEOREM 6. Let f : [,b] R be twice differentible function on,b such tht f L[,b]. If f is concve on [,b] for some fixed p 1 with = p p 1,nd f is liner mp, then the following ineulity for frctionl integrls holds for > : Γ 1 [ b fbjb ] b [ b 3 Q 1 f Q4 Q b } 3 1 Q 1 f Q4 bq b } } 3 Q 1 Q f Q5 Q b } 6 f Q5 bq b }] 6. 8 Where Q 1 = Q = Q 3 = Q 4 = Q 5 = = Q 1 t 1 = 1 Q 1 t 1 1 t 1 t 1 1 1 1 t = 1 3 t 1 t 1 = 1 3 1 t 1 1 t 1 t } t = 3 3 1 3 6 5. 6 } Q 6 = 1 t 1 1 t 1 t 1 t = 1 3 3 1 3 1 3 3
66 S. QAISAR, F. AHMAD, S. S. DRAGOMIR AND M. IQBAL Hence Proof. Using the concvity of f nd the power-men ineulity, we obtin f tx1 ty > t f x 1 t f y t f x 1 t f y f tx1 ty t f x 1 t f y, so, f is lso concve. By the Jensen integrl ineulity, we hve Γ 1 [ b fbjb ] b b 1 3 1 t 1 1 1 f 1 t1 t1 t b 1 t1 b 1 3 1 t 1 1 1 f 1 t1 tb1 t b 1 t1 b 1 3 1 t 1 1 t 1 t 1 1 f 1 t1 1 t 1 tt1 t b 1 t1 1 t 1 t b 1 3 1 t 1 1 t 1 t 1 1 f 1 t1 1 t 1 ttb1 t b 1 t1 1 t 1 t b 3 1 Q 1 f Q4 Q b 3 Q 1 b 3 1 Q 1 f Q4 bq b 3 Q 1 b 3 1 Q f Q5 Q b 6 Q b 3 1 Q f Q5 bq b 6 Q Γ 1 [ b fbj b ] b [ b 3 1 Q 1 f Q4 Q b } 3 Q 1 f Q4 bq b } } 3 Q 1 Q f Q5 Q b } 6 f Q5 bq b }] 6 Q Q
NEW RESULTS ON H-H INEQUALITY VIA FRACTIONAL INTEGRALS 663 The proof is completed. 1 b COROLLARY 3. On letting = 1 in Theorem 6, then ineulity 8 becomes s: b b fxdx b 48 [ 53b f 35b 8 f 8 ]. REMARK. Ineulity 9 is n improvement of obtined ineulity s in [11, Corollry 4] 9 Acknowledgements. The uthors would like to thnk the editor nd nonymous referee for the vluble comments nd suggestions on our mnsuscript. The uthors re grteful to Dr. S.M. Junid Zidi, Executive Director, COMSATS Institute of Informtion Technology, Pkistn for providing excellent reserch fcilities. Funding the uthor S. Qisr ws prtilly supported by the Higher Eduction Commission of Pkistn [Grnt Number. 535/Fedrl/NRPU/ R&D/HEC /16] R E F E R E N C E S [1] S. S. DRAGOMIR AND C. E. M. PEARCE, Selected topics on Hermite-Hdmrd ineulities nd pplictions, RGMIA Monogrphs, Victori University,, online: http://www.st.vu.edu.u/ RGMIA/monogrphs/hermite hdmrd.html. [] G. ANASTASSIOU, M. R. HOOSHMANDASL, A. GHASEMI, F. MOFTAKHARZADEH, Montogomery identities for frctionl integrls nd relted frctionl ineulities, J. Ine. Pure Appl. Mth. 1 4 9 Art. 97. [3] S. BELARBI, Z. DAHMANI, On some new frctionl integrl ineulities, J. Ine. Pure Appl. Mth. 1 3 9 Art. 86. [4] Z. DAHMANI, New ineulities in frctionl integrls, Interntionl Journl of Nonliner Science 9 4 1, 493 497. [5] Z. DAHMANI, On Minkowski nd Hermite-Hdmrd integrl ineulities vi frctionl integrtion, Ann. Funct. Anl. 1 1 1, 51 58. [6] Z. DAHMANI, L. TABHARIT, S. TAF, New generliztions of Grüss ineulity using Riemnn Liouville frctionl integrls, Bull. Mth. Anl. Appl. 3 1, 93 99. [7] Z. DAHMANI, L. TABHARIT, S. TAF, Some frctionl integrl ineulities, Nonl. Sci. Lett. A, 1 1, 155 16. [8] S. S. DRAGOMIR, M. I. BHATTI, M. IQBAL, AND M. MUDDASSAR, Some new frctionl Integrl Hermite-Hdmrd type ineulities, Journl of Computtionl Anlysis nd Appliction 18 4 15, 655 661. [9] M. IQBAL, S. QAISAR, M. MUDDASSAR, A short note on integrl ineulity of type Hermite- Hdmrd through convexity, Journl of computtionl nlysis nd ppliction 1 5 16, 946 953. [1] M. I. BAHTTI, M. IQBAL, S. S. DRAGOMIR, Some new frctionl integrl ineulities Hermite- Hdmrd type ineulities, Journl of computtionl nlysis nd ppliction 16 4 15, 643 653. [11] M. E. ÖZDEMIR, C. YILDIZ, A. O. AKDEMIR, E. SET, On some ineulities for s-convex functions nd pplictions, J. Ine. & Appl. 13 333. [1] S. QAISAR, M. IQBAL, M. MUDDASSAR, New Hermite-Hdmrd s ineulities for preinvex function vi frctionl integrls, Journl of computtionl nlysis nd ppliction 7 16, 1318 138.
664 S. QAISAR, F. AHMAD, S. S. DRAGOMIR AND M. IQBAL [13] M. Z. SARIKAYA, E. SET, H. YALDIZ AND N. BASAK, Hermite-Hdmrd s ineulities for frctionl integrls nd relted frctionl ineulities, Mthemticl nd Computer Modelling 57 9 1, 43 47, 13. [14] M. Z. SARIKAYA, N. AKTAN, On the generliztion of some integrl ineulities nd their pplictions, Mthemticl nd computer Modelling 54 9 11, 175 18. [15] C. FEIXIANG, Extension of the Hermite-Hdmrd ineulity for convex function vi frctionl integrls, Journl of Mthemticl Ineulities 1 1 16, 75 81. Received December 6, 16 Shhid Qisr Comsts Institute of Informtion Technology Shiwl Pkistn e-mil: shhidisr9@ciitshiwl.edu.pk Froo Ahmd Punjb Higher Eduction Deprtment Den Fculty of Sciences, Govt. Islmi College Civil Lines Lhore, Punjb, 54, Pkistn e-mil: froogujjr@gmil.com; froogujr@giccl.edu.pk S. S. Drgomir School of Engineering nd Science Victori University P. O. Box 1448 Melbourne City, VIC 81 Austrli e-mil: sever.drgomir@vu.edu.u Muhmmd Ibl University of Engineering nd Technology Lhore, Pkistn e-mil: ibl uet68@yhoo.com Journl of Mthemticl Ineulities www.ele-mth.com jmi@ele-mth.com