Interpolating inequalities for functions of positive semidefinite matrices

Abstract

Let $A$, $B$ be positive semidefinite $n\times n$ matrices, and let $\alpha \in (0,1)$. We show that if $f$ is an increasing submultiplicative function on $[0,\infty )$ with $f\left(0\right)=0$ such that $f\left(t\right)$ and $f^2\left(t^{1/2}\right)$ are convex, then for every unitarily invariant norm. Moreover, if $\alpha \in [0,1]$ and $X$ is an $n\times n$ matrix with $X\ne 0$, then for every unitarily invariant norm. These inequalities present generalizations of recent results of Zou and Jiang and of Audenaert.