# A note on some inequalities for positive linear maps

## TL;DR

This paper improves and generalizes inequalities involving positive linear maps and operators, providing tighter bounds and an enhanced operator Pólya-Szegö inequality for specific operator orderings.

## Contribution

It introduces new bounds for operator inequalities under certain orderings and extends the operator Pólya-Szegö inequality with improved constants.

## Key findings

- Derived new inequalities for positive linear maps with explicit bounds.
- Generalized the operator Pólya-Szegö inequality with improved constants.
- Applicable to operators within specific orderings and bounds.

## Abstract

We improve and generalize some operator inequalities for positive linear maps. It is shown, among other inequalities, that if $0<m\le B\le m'<M'\le A\le M$ 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.