Improved Jensen-type inequalities via linear interpolation and applications

TL;DR

This paper introduces a novel method using linear interpolation to enhance Jensen-type inequalities for convex and piecewise convex functions, leading to improved inequalities in both scalar and matrix contexts.

Contribution

The paper presents a new general approach for refining Jensen-type inequalities through linear interpolation, applicable to scalar and matrix inequalities, advancing existing mathematical bounds.

Findings

Improved bounds for Young-type inequalities.

Enhanced matrix inequalities using the proposed method.

Applicability to piecewise convex functions.

Abstract

In this paper we develop a general method for improving Jensen-type inequalities for convex and, even more generally, for piecewise convex functions. Our main result relies on the linear interpolation of a convex function. As a consequence, we obtain improvements of some recently established Young-type inequalities. Finally, our method is also applied to matrix case. In such a way we obtain improvements of some important matrix inequalities known from the literature.