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}$n}X\sharp_t Z d\mu(Z)$\mathrm{.}Inthispaper,weshowthat$$ P_t(\mu)\leq I\quad \Longrightarrow\quad P{\frac{t}{p}}(\nu)\leq P_t(\mu)$$forevery$p\geq1$,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 $\mathrm{\Phi }:{\mathbb{M}}_n\to {\mathbb{M}}_m$ is a unital positive linear map, then $\mathrm{\Phi }(P_t(\mu ))\le P_t(\nu )$ for all $t\in [-1,1]\mathrm{\backslash }\{0\}$, where $\nu$ is a certain measure.