A glimpse at the operator Kantorovich inequality

Abstract

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

to appear in Linear Multilinear Algebra