Some properties of h-MN-convexity and Jensen's type inequalities

In this work, we introduce the class of h-MN-convex functions by generalizing the concept of MN-convexity and combining it with h-convexity. Namely, Let I, J be two intervals subset of (0, infinity) such that (0,1) subset of J and [a, b] subset of I. Consider a non-negative function h : (0, infinity) -> (0, infinity) and let M : [0, 1] -> [a, b] (0 < a < b) be a Mean function given by M(t) = M(h(t); a, b); where by M(h(t); a, b) we mean one of the following functions: A(h)(a, b) := h(1 - t) a + h(t) b, G(h)(a, b) = a(h(1-t)) b(h(t)) and H-h(a,b) := ab/h(t)a+h(1-t)b = 1/A(h)(1/a,1/b); with the property that M(h(0); a, b) and M(h(1); a, b) = b. A function f : I -> (0, infinity) is said to be h-MN-convex (concave) if the inequality f(M(t; x, y)) <= (>=) N(h(t); f (x), f (y)), holds for all x, y is an element of I and t is an element of [0,1], where M and N are two mean functions. In this way, nine classes of h-MN-convex functions are established and some of their analytic properties are explored and investigated. Characterizations of each type are given. Various jensen's type inequalities and their converses are proved.