# Various inequalities between quasi-arithmetic mean and quasi-geometric type means for matrices

## TL;DR

This paper investigates inequalities between various quasi-arithmetic and quasi-geometric means of positive semidefinite matrices, establishing conditions on parameters for these inequalities to hold under different matrix orderings.

## Contribution

It introduces new inequalities between matrix means and determines necessary and sufficient conditions for these inequalities across different matrix orderings.

## Key findings

- Derived conditions for inequalities between matrix means.
- Extended classical inequalities to matrix settings.
- Analyzed inequalities under multiple matrix orderings.

## Abstract

In this paper, for $0<\alpha<1$, $p>0$ and positive semidefinite matrices $A,B\ge0$, we consider the quasi-extension $\mathcal{A}_{\alpha,p}(A,B):=((1-\alpha)A^p+\alpha B^p)^{1/p}$ of the $\alpha$-weighted arithmetic matrix mean, and the quasi-extensions $\mathcal{M}_{\alpha,p}(A,B):=\mathcal{M}_\alpha(A^p,B^p)^{1/p}$ of several different $\alpha$-weighted geometric-type matrix means $\mathcal{M}_\alpha(A,B)$ such as the $\alpha$-weighted geometric mean in Kubo and Ando's sense and two types of $\alpha$-weighted version of Fiedler and Pt\'ak's spectral geometric mean, as well as the R\'enyi mean and the $\alpha$-weighted Log-Euclidean mean. For these we examine the inequalities $\mathcal{A}_{\alpha,p}(A,B)\triangleleft\mathcal{A}_{\alpha,q}(A,B)$ and $\mathcal{M}_{\alpha,p}(A,B)\triangleleft\mathcal{A}_{\alpha,q}(A,B)$ of arithmetic-geometric type, where $\triangleleft$ is one of several different matrix orderings varying from the strongest Loewner order to the weakest order determined by trace inequality. For each choice of the above inequalities, our goal is to hopefully obtain the necessary and sufficient condition on $p,q,\alpha$ under which the inequality holds for all $A,B\ge0$.