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 ≤ A ≤ M . Then for every unital positive linear map Φ , Φ ( A − 1 ) 2 ≤ ( ( M + m ) 2 4 M m ) 2 Φ ( A ) − 2 . As a consequence, Φ ( A − 1 ) Φ ( A ) + Φ ( A ) Φ ( A − 1 ) ≤ ( M + m ) 2 2 M m .