# Some inequalities of matrix power and Karcher means for positive linear maps

## TL;DR

This paper extends matrix inequalities involving matrix power and Karcher means for positive definite matrices, providing bounds under certain conditions with applications to positive linear maps.

## Contribution

It generalizes existing inequalities for matrix power and Karcher means, introducing new bounds involving positive unital linear maps and parameters.

## Key findings

- Derived new inequalities for matrix power and Karcher means.
- Established bounds involving positive unital linear maps.
- Provided conditions under which inequalities hold.

## Abstract

In this paper, we generalize some matrix inequalities involving matrix power and Karcher means of positive definite matrices. Among other inequalities, it is shown that if ${\mathbb A}=(A_{1},...,A_{n})$ is a $n$-tuple of positive definite matrices such that $0<m\leq A_{i}\leq M\, (i=1,\cdots,n)$ for some scalars $m< M$ and $\omega=(w_{1},\cdots,w_{n})$ is a weight vector with $w_{i}\geq0$ and $\sum_{i=1}^{n}w_{i}=1$, then \begin{align*} \Phi^{p}\Big(\sum_{i=1}^{n}w_{i}A_{i}\Big)\leq \alpha^{p}\Phi^{p}(P_{t}(\omega; {\mathbb A})) \end{align*} and \begin{align*} \Phi^{p}\Big(\sum_{i=1}^{n}w_{i}A_{i}\Big)\leq \alpha^{p}\Phi^{p}(\Lambda(\omega; {\mathbb A})), \end{align*} where $p>0$, $\alpha=\max\Big\{\frac{(M+m)^{2}}{4Mm}, \frac{(M+m)^{2}}{4^{\frac{2}{p}}Mm}\Big\}$, $\Phi$ is a positive unital linear map and $t\in [-1, 1]\backslash \{0\}$.