On a reverse of Ando--Hiai inequality

Abstract

In this paper, we show a complement of Ando--Hiai inequality: Let $A$ and $B$ be positive invertible operators on a Hilbert space $H$ and $\alpha\in [0,1]$. If $A\ \sharp_{\alpha}\ B \leq I$, then $$A^r\ \sharp_{\alpha} \ B^r \leq \| (A\ \sharp_{\alpha}\ B)^{-1} \| ^{1-r} I {for all positive number $r\leq 1$,$$ where $I$ is the identity operator and the symbol $\| \cdot \|$ stands for the operator norm.