Convex functions and means of matrices

TL;DR

This paper explores refined inequalities for matrices involving convex and log-convex functions, improving classical results like Young's and Heinz inequalities through new mathematical refinements.

Contribution

It introduces novel refinements and reverses of classical matrix inequalities using properties of convex and log-convex functions.

Findings

Refined versions of Young's inequality for matrices

Enhanced Heinz inequality bounds

Reverses of arithmetic-harmonic and geometric-harmonic mean inequalities

Abstract

In this article, we prove that convex functions and log-convex functions obey certain general refinements that lead to several refinements and reverses of well known inequalities for matrices, including Young's inequality, Heinz inequality, the arithmetic-harmonic and the geometric-harmonic mean inequalities.