Power means of probability measures and Ando-Hiai inequality

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