Hermite-Hadamard inequalities for differentiable p-convex functions using hypergeometric functions

Recently theory of convexity has received much attentions by many researchers. Consequently the classical concepts of convex sets and convex functions have been extended and generalized in several directions using novel and innovative ideas, see [1]. Zhang [11] introduced the notion of p-convex functions. It is worth to mention here that besides the classical convex functions, the class of p-convex functions also includes the class of harmonically convex functions introduced and studied by Iscan [5]. For some recent investigations on p-convex functions, see [4]. The interrelationship between theory of convex functions and theory of inequalities has attracted many researchers. One of the most extensively studied inequality for convex functions is the Hermite–Hadamard inequality. This inequality provides the necessary and sufficient condition for a function to be convex. For some recent investigation on Hermite–Hadamard type inequalities, see [2–10]. In this article, We consider the class of p-convex functions. We derive two new integral identities for differentiable functions. Using these results we establish our main results that are Hermite–Hadamard type inequalities for differentiable p-convex functions. We use hypergeometric functions to solve our integrals. It is expected that the ideas and techniques of this paper may stimulate further research in this area. This is the main motivation of this paper.