Two trace inequalities for operator functions

Abstract

In this paper we show that for a non-negative operator monotone function $f$ on $[0,\infty )$ such that $f(0)=0$ and for any positive semidefinite matrices $A$ and $B$, $$Tr((A-B)(f(A)-f(B)))\le Tr(|A-B|f(|A-B|)).$$ When the function $f$ is operator convex on $[0,\infty )$, the inequality is reversed.

7 pages, final version, to be published in MIA (2019)