Some integral inequalities for harmonic h-convex functions involving hypergeometric functions

The aim of this paper is to establish some new Hermite–Hadamard type inequalities for harmonic h-convex functions involving hypergeometric functions. We also discuss some new and known special cases, which can be deduced from our results. The ideas and techniques of this paper may inspire further research in this field. In recent years, much attention have been given to theory of convexity because of its great utility in various fields of pure and applied sciences. Many researchers have extended and generalized the classical concepts of convex sets and convex functions in various directions using novel and innovative techniques. For more information, see [1–4,6,9,14–17,19]. To unify the classes of classical convex functions, s-Breckner convex functions [1], Godunova–Levin functions [6] and P-functions [4], Varošanec [19] introduced the concept of h-convex functions. Is ßcan [9] introduced another new class of convex functions which is called harmonically convex functions. For some recent investigations on harmonically convex functions, see [5,18]. Noor et al. [16] introduced the concept of harmonically h-convex functions, which generalizes several new and known class of harmonically convex functions. A very interested inequality associated with convex functions is called the Hermite–Hadamard type inequality. This inequality provides a necessary and sufficient condition for a function to be convex. f a þ b 2 6 1 b À a Z b a f ðxÞdx 6 f ðaÞ þ f ðbÞ 2 ð1:1Þ holds if and only if f is convex. The inequality (1.1) has been extended and generalized for various classes of convex functions via different approaches, see [3–5,7,9–13,15–18]. We derive some new Hermite–Hadamard type inequalities for harmonically h-convex functions. Results proved continue to hold for various known and new classes of convex functions. It is expected that the ideas and techniques of this paper may stimulate further research in this field.