New inequalities for operator concave functions involving positive linear maps

TL;DR

This paper introduces new inequalities for operator concave functions involving positive linear maps, extending the operator Kantorovich inequality and unifying known inequalities under a broader framework.

Contribution

It presents general inequalities for operator concave functions involving positive linear maps, including extensions of the operator Kantorovich inequality and known results.

Findings

Derived new inequalities for operator concave functions.

Extended the operator Kantorovich inequality.

Unified existing inequalities as special cases.

Abstract

The purpose of this paper is to present some general inequalities for operator concave functions which include some known inequalities as a particular case. Among other things, we prove that if $A\in \mathcal{B}\left( \mathcal{H} \right)$ is a positive operator such that $mI\le A\le MI$ for some scalars $0<m<M$ and $\Phi $ is a normalized positive linear map on $\mathcal{B}\left( \mathcal{H} \right)$, then \[\begin{aligned} {{\left( \frac{M+m}{2\sqrt{Mm}} \right)}^{r}}&\ge {{\left( \frac{\frac{1}{\sqrt{Mm}}\Phi \left( A \right)+\sqrt{Mm}\Phi \left( {{A}^{-1}} \right)}{2} \right)}^{r}} & \ge \frac{\frac{1}{{{\left( Mm \right)}^{\frac{r}{2}}}}\Phi {{\left( A \right)}^{r}}+{{\left( Mm \right)}^{\frac{r}{2}}}\Phi {{\left( {{A}^{-1}} \right)}^{r}}}{2} & \ge \Phi {{\left( A \right)}^{r}}\sharp\Phi {{\left( {{A}^{-1}} \right)}^{r}}, \end{aligned}\] where $0\le r\le 1$, which nicely extend the operator Kantorovich inequality.