  We show the following result: Let $A$ be a positive operator satisfying
$0<m{{\mathbf{1}}_{\mathcal{H}}}\le A\le M{{\mathbf{1}}_{\mathcal{H}}}$ for
some scalars $m,M$ with $m<M$ and $\Phi $ be a normalized positive linear map,
then \[\Phi \left( {{A}^{-1}} \right)\le \Phi \left(
{{m}^{\frac{A-M{{\mathbf{1}}_{\mathcal{H}}}}{M-m}}}{{M}^{\frac{m{{\mathbf{1}}_{\mathcal{H}}}-A}{M-m}}}
\right)\le \frac{{{\left( M+m \right)}^{2}}}{4Mm}\Phi {{\left( A
\right)}^{-1}}.\]
