A note on some inequalities for positive linear maps

Abstract

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

to appear in Linear Multilinear Algebra