Trace inequalities for completely monotone functions and Bernstein functions

TL;DR

This paper establishes new matrix trace inequalities for completely monotone and Bernstein functions, including special cases that enhance or complement existing inequalities like McCarthy's, advancing the theoretical understanding of these functions.

Contribution

It introduces novel trace inequalities for classes of functions, notably for power functions, that extend and strengthen prior results such as McCarthy's inequality.

Findings

Derived matrix trace inequalities for completely monotone functions

Established trace inequalities for Bernstein functions

Extended McCarthy's inequality with new bounds

Abstract

We prove a matrix trace inequality for completely monotone functions and for Bernstein functions. As special cases we obtain non-trivial trace inequalities for the power function x->x^q, which for certain values of q complement McCarthy's trace inequality and for others strenghten it.