Reverses and Refinements of Jensen's Inequality for Positive Linear Functionals on Hermitian Unital Banach *-Algebras

TL;DR

This paper develops new inequalities and refinements related to Jensen's inequality for positive linear functionals on Hermitian unital Banach *-algebras, with applications to specific convex functions.

Contribution

It introduces reverses and refinements of Jensen's and Slater's inequalities in the context of Hermitian unital Banach *-algebras, expanding the theoretical framework.

Findings

New inequalities for convex functions on Banach *-algebras

Reverses and refinements of Jensen's and Slater's inequalities

Examples illustrating the inequalities for particular convex functions

Abstract

We establish in this paper some inequalities for analytic and convex functions on an open interval and positive normalized functionals defined on a Hermitian unital Banach *-algebra. Reverses and refinements of Jensen's and Slater's type inequalities are provided. Some examples for particular convex functions of interest are given as well.