We show the following result: Let $A,B\in \mathbb{ B}\left( \mathcal{ H} \right)$ be two strictly positive operators such that $A\le B$ and $m{ { \mathbf{ 1}}_{ \mathcal{ H}}}\le B\le M{ { \mathbf{ 1}}_{ \mathcal{ H}}}$ for some scalars $0<m<M$. Then \[{ { B}^{ p}}\le \exp \left( \frac{ M{ { \mathbf{ 1}}_{ \mathcal{ H}}}-B}{ M-m}\ln { { m}^{ p}}+\frac{ B-m{ { \mathbf{ 1}}_{ \mathcal{ H}}}}{ M-m}\ln { { M}^{ p}} \right)\le K\left( m,M,p,q \right){ { A}^{ q}}\quad\text{ for }p\le 0,-1\le q\le 0\] where $K\left( m,M,p,q \right)$ is the generalized Kantorovich constant with two parameters. In addition, we obtain Kantorovich type inequalities for the chaotic order.
Categories: math.FA