Some Matrix Power and Karcher Means Inequalities Involving Positive Linear Maps

Abstract

In this paper, we generalize some matrix inequalities involving the matrix power means and Karcher mean of positive definite matrices. Among other inequalities, it is shown that if
${\mathbb A}=(A_{1},...,A_{n})$ is an $n$-tuple of positive definite matrices such that  $0<m\leq A_{i}\leq M(i=1,...,n)$ for some scalars $m< M$ and $\omega=(w_{1},...,w_{n})$ is a weight vector with $w_{i}\geq0$ and $\sum_{i=1}^{n}w_{i}=1$, then
\begin{align*}
\Phi^{p}\left(\sum_{i=1}^{n}w_{i}A_{i}\right)\leq \alpha^{p}\Phi^{p}(P_{t}(\omega; {\mathbb A}))
\end{align*}
and
\begin{align*}
\Phi^{p}\left(\sum_{i=1}^{n}w_{i}A_{i}\right)\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  $t\in [-1, 1]\backslash \{0\}$.