  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.
