Power means of probability measures and Ando-Hiai inequality

Abstract

Let $μ$ be a probability measure of compact support on the set $\mathbb{P}_n$ of all positive definite matrices, let $t\in(0,1]$, and let $P_t(μ)$ be the unique positive solution of $X=\int_{\mathbb{P}_n}X\sharp_t Z dμ(Z)$. In this paper, we show that $$ P_t(μ)\leq I\quad \Longrightarrow\quad P_{\frac{t}{p}}(ν)\leq P_t(μ)$$ for every $p\geq1$, where $ν(Z)=μ(Z^{1/p})$. This provides an extension of the Ando--Hiai inequality for matrix power means. Moreover, we prove that if $Φ:\mathbb{M}_n\to\mathbb{M}_m$ is a unital positive linear map, then $Φ(P_t(μ))\leq P_t(ν)$ for all $t\in[-1,1]\backslash\{0\}$, where $ν$ is a certain measure.