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 $\Phi$, \[\Phi(A^{-1})^2\le (\frac{(M+m)^2}{4Mm})^2\Phi(A)^{-2}.\] As a consequence, \[\Phi(A^{-1})\Phi(A)+\Phi(A)\Phi(A^{-1}) \le \frac{(M+m)^2}{2Mm}.\]