Improvements of Jensen's inequality and its converse for strongly convex functions with applications to strongly f-divergences

Abstract

In this paper, using the class of strongly convex functions, which is subclass of convex functions with stronger versions of analogous properties, we get improvements of Jessen's and Jensen's inequalities, their converses as well as related Jensen-type interpolating inequalities which present a starting point in many significant results in recent investigations. Obtained improvements we apply to so called strongly f-divergences, a concept of f-divergences for strongly convex functions. As outcome we derive stronger estimates for some well known divergences as the Kullback-Leibler divergence, χ2-divergence, Hellinger divergence, Bhattacharya distance and Jeffreys distance.