  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 $t\in [-1, 1]\backslash \{0\}$.
