Matrix Hermite-Hadamard type inequalities

TL;DR

This paper develops new matrix and operator inequalities of Hermite-Hadamard type, extending classical inequalities to the matrix and operator setting with applications in functional analysis.

Contribution

It introduces majorization and operator versions of Hermite-Hadamard inequalities for convex functions on matrices and operators, using the Mond-Pecaric method.

Findings

Established a majorization version for monotone convex functions on matrices

Derived an operator version for convex functions using the Mond-Pecaric method

Presented applications and an Hermite-Hadamard inequality for operator convex functions

Abstract

We present several matrix and operator inequalities of Hermite-Hadamard type. We first establish a majorization version for monotone convex functions on matrices. We then utilize the Mond-Pecaric method to get an operator version for convex functions. We also present some applications. Finally we obtain an Hermite-Hadamard inequality for operator convex functions, positive linear maps and operators acting on Hilbert spaces.