Some notes on Jensen-Mercer's type inequalities; extensions and refinements with applications

Abstract

. In this paper we study inequalities corresponding to Jensen-Mercer’s inequality. Some new extensions of Niezgoda’s inequality and the integral version of Jensen-Mercer’s inequality are given. The obtained inequalities do not only generalize the former ones, but our proofs are natural and simple. They clearly show the structure of such inequalities: they consist of two parts, a discrete or integral Jensen’s inequality and then a majorization type inequality. An- other purpose of the paper is to provide a deeper understanding of the methods used to re fi ne Jensen-Mercer’s and the corresponding inequalities. Moreover, some new re fi nements of these inequalities are obtained. Finally, some applications related to Fej´er’s and Hermite-Hadamard inequalities are given.