On uniformly GA-convex functions

Abstract

An ordinary convexity can be interpreted in the form of an inequality between arithmetic means and referred as to AA-convexity. Other classes of convex functions that include means are also known in the literature. Depending on which type of mean is included, arithmetic A or geometric G, there are also GG-convex, AG-convex and GA-convex functions. On the other side, a class with stronger property that ordinary convex class is known as uniform convexity. In this paper, we connect these two concepts, GA-convexity with the uniform convexity, and introduce a new concept named uniform GA-convexity. By analyzing the newly defined class we prove that it inherits some good properties from both classes of convexity. For uniformly GA-convex functions we prove few basic inequalities as Jensen's inequality, the Jensen-Mercer inequality and the Hermite-Hadamard inequality. As applications of the main results we obtain some analytic inequalities and new estimates of some statistical divergences.