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

Submitted on 24 November 2018