  We improve and generalize some operator inequalities for positive linear
maps. It is shown, among other inequalities, that if $0<m\le B\le m'<M'\le A\le
M$ or $0<m\le A\le m'<M'\le B\le M$, then for each $2\le p<\infty $ and $\nu
\in \left[ 0,1 \right]$, \begin{equation*} {{\Phi }^{p}}\left( A{{\nabla }_{\nu
}}B \right)\le {{\left( \frac{K\left( h
\right)}{{{4}^{\frac{2}{p}-1}}{{K}^{r}}\left( h' \right)} \right)}^{p}}{{\Phi
}^{p}}\left( A{{\#}_{\nu }}B \right), \end{equation*} and \begin{equation*}
{{\Phi }^{p}}\left( A{{\nabla }_{\nu }}B \right)\le {{\left( \frac{K\left( h
\right)}{{{4}^{\frac{2}{p}-1}}{{K}^{r}}\left( h' \right)} \right)}^{p}}{{\left(
\Phi \left( A \right){{\#}_{\nu }}\Phi \left( B \right) \right)}^{p}},
\end{equation*} where $r=\min \left\{ \nu ,1-\nu \right\}$, $h=\frac{M}{m}$ and
$h'=\frac{M'}{m'}$. We also obtain an improvement of operator P\'olya-Szeg\"o
inequality.
