New inequalities for operator concave functions involving positive linear maps

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.