### 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  $t\in [-1, 1]\backslash \{0\}$.

PDF

### Refbacks

• There are currently no refbacks.