HERMITE--HADAMARD TYPE INEQUALITIES VIA DIFFERENTIABLE hφ−h_{\varphi }-hφ− PREINVEX FUNCTIONS FOR FRACTIONAL INTEGRALS

Year 2021, Volume: 4 Issue: 1, 87 - 101, 31.01.2021

Abdullah Akkurt , Hüseyin Yıldırım

https://izlik.org/JA84KA49JF

Abstract

In this paper, we consider a new class of convex functions which is called h_varphi-preinvex functions. We prove several Hermite--Hadamard type

inequalities for differentiable h_varphi-preinvex functions via Fractional Integrals. Some special cases are also discussed. Our results

extend and improve the corresponding ones in the literature.

Keywords

integral inequalities , fractional integrals , Hermite-Hadamard Inequality

References

• Barani, A.; Ghazanfari, A.G.; Dragomir, S.S.: Hermite-Hadamardinequality for functions whose derivatives absolute values are preinvex. J.Inequal. Appl. 2012, 247 (2012)
• Dragomir, S.S.; Pearce, C.E.M.: Selected topics onHermite--Hadamard inequalities and applications. Victoria University (2000)
• Dragomir, S.S.; Pecaric, J.; Persson, L.E.: Some inequalities ofHadamard type. Soochow J. Math. 21, 335--341, (1995)
• \.{I}\c{s}can, \.{I}.: Hermite--Hadamard's inequalities forpreinvex function via fractional integrals and related functionalinequalities, Am. J. Math. Anal. 1 (3) (2013) 33--38.
• Latif, M.A.: Some inequalities for differentiable prequasiinvexfunctions with applications. Konuralp J. Math. 1(2), 17--29 (2013)
• Latif, M.A.; Dragomir, S.S.: Some Hermite-Hadamard typeinequalities for functions whose partial derivatives in absloute value arepreinvex on the co-oordinates. Facta Universitatis (NIS) Ser. Math. Inform.28(3), 257--270, (2013)
• Latif, M.A.; Dragomir, S. S.; Momoniat, E.: Some weightedintegral inequalities for differentiable preinvex and prequasiinvexfunctions. RGMIA (2014)
• Noor, M.A.: On Hadamard integral inequalities involving twolog-preinvex functions. J. Inequal. Pure Appl. Math. 8, 1--6 (2007)
• Noor, M.A.: On Hermite-Hadamard integral inequalities forproduct of two nonconvex functions. J. Adv. Math. Studies 2(1), 53--62 (2009)
• Noor, M.A.: Some new classes of nonconvex functions. Nonl.Funct. Anal. Appl. 11(1), 165--171 (2006)
• Noor, M.A.; Awan, M.U.; Noor, K.I.: On some inequalities forrelative semi-convex functions. J. Inequal. Appl., 2013, 332 (2013)
• Noor, M.A.; Noor, K.I.: Generalized preinvex functions andtheir properties. Journal of Appl. Math. Stochastic Anal., 2006(12736),1--13, doi:10.1155/JAMSA/2006/12736
• Noor, M.A.; Noor, K.I.; Ashraf, M.A.; Awan, M.U.; Bashir, B.:Hermite--Hadamard inequalities for $h_{\varphi }-$convex functions. Nonl.Anal. Formum. 18, 65--76 (2013)
• Noor, M.A.; Noor, K.I.; Awan, M.U.: Hermite--Hadamardinequalities for relative semi-convex functions and applications. Filomat,28(2), 221--230, (2014)
• Noor, M.A.; Noor, K.I.; Awan, M.U.: Generalized convexity andintegral inequalities. Appl. Math. Inf. Sci. 9(1), 233--243, (2015)
• Noor, M.A.; Noor, K.I.; Awan, M.U.; Khan, S.: FractionalHermite-Hadamard inequalities for some new classes of Godunova--Levinfunctions. Appl. Math. Inf. Sci. 8(6), 2865--2872, (2014)
• Noor, M.A.; Noor, K.I.; Awan, M.U.; Li, J.: OnHermite--Hadamard type Inequalities for h-preinvex functions, Filomat 28:7(2014), 1463--1474, DOI 10.2298/FIL1407463N
• Noor, M.A.; Noor, K.I.; Al-Said, E.: Iterative methods forsolving nonconvex equilibrium variational inequalities. Appl. Math. Inf.Sci. 6(1), 65--69, (2012).
• Noor, M.A.; Postolache, M.; Noor, K.I.; Awan, M.U.:Geometrically nonconvex functions and integral inequalities. Appl. Math.Inf. Sci. 9, (2015).
• Noor, M.A.; Noor, K.I.; Awan, M.A.; Khan, S.: Hermite--Hadamardtype inequalities for differantiable $h_{\varphi }-$preinvex functions.Arab. J. Math.4:63-76 (2015).
• Sarikaya, M.Z.; Set, E.: \"{O}zdemir M.E.: On some newinequalities of Hadamard type involving h-convex functions. Acta Math. Univ.Comenianae. 2, 265--272 (2010).
• Sarikaya, M.Z.; Alp, N.; Bozkurt, H.: On Hermite-Hadamard TypeIntegral Inequalities for preinvex and log-preinvex functions, ContemporaryAnalysis and Applied Mathematics, Vol.1, No.2, 237-252, 2013.
• Varosanec, S.: On h-convexity. J. Math. Anal. Appl. 326,303--311 (2007)
• Weir, T.; Mond, B.: Preinvex functions in multiple objectiveoptimization. J. Math. Anal. Appl. 136, 29--38 (1998)
• Samko, S.G.; Kilbas, A.A.; Marichev, O.I.: Fractional Integralsand Derivatives, Theory and Applications, Gordon and Breach, Yverdon,Switzerland, 1993