On an operator Kantorovich inequality for positive linear maps

Abstract

We improve the operator Kantorovich inequality as follows: Let $A$ be a positive operator on a Hilbert space with $0<m\le A\le M$. Then for every unital positive linear map $\Phi$, \[Φ(A^{-1})^2\le (\frac{(M+m)^2}{4Mm})^2Φ(A)^{-2}.\] As a consequence, \[Φ(A^{-1})Φ(A)+Φ(A)Φ(A^{-1}) \le \frac{(M+m)^2}{2Mm}.\]