Power means of probability measures and Ando-Hiai inequality

TL;DR

This paper extends the Ando-Hiai inequality to matrix power means of probability measures, establishing new inequalities and properties involving positive linear maps and matrix power means.

Contribution

The paper introduces an extension of the Ando-Hiai inequality for matrix power means of probability measures, including new bounds and properties involving positive linear maps.

Findings

Extended Ando-Hiai inequality for matrix power means.

Established inequalities involving positive linear maps.

Provided conditions under which inequalities hold for measures and maps.

Abstract

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.