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 $$A^r \, \sharp_\alpha \, B^r \leq \|(A \,\sharp_\alpha \, B)^{-1}\|^{1-r} I \quad \text{for all }\, 0 <r \leq 1, $$ where $I$ is the identity operator and the symbol $\| \cdot \|$ stands for the operator norm.