TJMM 6 (2014), No. 2, 171-180 HERMITE-HADAMARD INEQUALITIES FOR MODIFIED h-CONVEX FUNCTIONS MUHAMMAD ASLAM NOOR, KHALIDA INAYAT NOOR, AND MUHAMMAD UZAIR AWAN Abstract. In this paper, we consider the class of modified h-convex functions, which was introduced by Toader [14]. We derive Hermite-Hadamard type inequalities for the modified h-convex functions. Some special cases are also discussed. We try to show that this class enjoys some nice properties which the convex functions have. 1. Introduction Convexity plays a pivotal role in different fields of applied and pure sciences. In recent years, several extensions and generalizations of the concept of convexity have been con- sidered using some novel and innovative techniques, see [2, 4, 5, 7, 8, 12, 13, 14, 15, 16]. Varoˇ sanec [15] introduced a significant class of nonconvex function, which is called h- convex function. She has noticed that this class generalizes several other classes of non- convex functions along with classical convexity. For some recent studies on h-convex functions, see [9, 10, 11]. Toader [14] introduced a new class of nonconvex functions, which is called as (h, λ, µ)-convex functions. Toader studied the basic properties for this class of nonconvex functions. Let f : I ⊆ R → R be a convex function with a<b and a, b ∈ I . Then the following double inequality is known as Hermite-Hadamard inequality in the literature f  a + b 2  ≤ 1 b − a b  a f (x)dx ≤ f (a)+ f (b) 2 . named after C. Hermite and J. Hadamard. This result can be considered as a necessary and sufficient condition for a function to be convex. Interested readers are referred to [1] for useful details on Hermite-Hadamard inequalities and its variant forms. Fej´ er [3], had given a generalization of the Hermite-Hadamard inequality as, if f : [a, b] → R is a convex function and g :[a, b] → R is nonnegative, integrable and symmetric about a+b 2 , then f  a + b 2  b  a w(x)dx ≤ b  a f (x)w(x)dx ≤ f (a)+ f (b) 2 b  a w(x)dx. For some recent investigations on Hermite-Hadamard type inequalities and on its vari- ant forms, see [1, 2, 3, 7, 8, 9, 10, 11]. 2010 Mathematics Subject Classification. 26D15; 26A51. Key words and phrases. Convex functions, h-convex functions, s-convex, P -convex, Godunova-Levin functions, α-convex functions, Hermite-Hadamard inequality. The authors are grateful to Dr. S. M. Junaid Zaidi, Rector, COMSATS Institute of Information Technology, Pakistan, for providing excellent research facilities. The authors also would like to thank Prof. Dr. G. Toader for his valuable comments and remarks on the earlier version of the manuscript. This research is supported by HEC NRPU project No: 20-1966/R&D/11-2553. 171