Matrix Fej\'{e}r and Levin-Ste\v{c}kin Inequalities

TL;DR

This paper extends classical Fejér and Levin-Steckin inequalities to matrix settings, providing majorization results for convex and log-convex functions and matrix inequalities involving Hermitian matrices.

Contribution

It introduces matrix versions of Fejér and Levin-Steckin inequalities, including majorization results and partial Loewner ordering for Hermitian matrices.

Findings

Matrix Fejér-type majorization results for convex functions

Matrix Levin-Steckin inequalities involving Loewner ordering

Rigorous results for Hermitian matrices

Abstract

Fe{j}\'er and Levin-Ste\v{c}kin inequalities treat integrals of the product of convex functions with symmetric functions. The main goal of this article is to present possible matrix versions of these inequalities. In particular, majorization results are shown of Fej\'{e}r type for both convex and log-convex functions. For matrix Levin-Ste\v{c}kin type, we present more rigorous results involving the partial Loewner ordering for Hermitian matrices.