Sherman's inequality and its converse for strongly convex functions with applications to generalized f-divergences

TL;DR

This paper extends Sherman's inequality to strongly convex functions, introduces a converse inequality, and applies these results to generalized f-divergences, providing stronger versions of classical inequalities.

Contribution

The paper generalizes Sherman's inequality to strongly convex functions and develops a converse inequality, enhancing the theoretical framework for inequalities involving strongly convex functions.

Findings

Extended Sherman's inequality to strongly convex functions

Derived a converse Sherman inequality with upper bounds

Applied results to generalized f-divergences and reverse relations

Abstract

Considering the weighted concept of majorization, Sherman obtained generalization of majorization inequality for convex functions known as Sherman's inequality. We extend Sherman's result to the class of n-strongly convex functions using extended idea of convexity to the class of strongly convex functions. We also obtaine upper bound for Sherman's inequality, so called the converse Sherman inequality, and as easy consequences we get Jensen's as well as majorization inequality and their conversions for strongly convex functions. Obtained results are stronger versions for analogous results for convex functions. As applications, we introduced a generalized concept of f-divergence and derived some reverse relations for such concept.