On an operator Kantorovich inequality for positive linear maps

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

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