New Inequalities of the Kantorovich Type With Two Negative Parameters

Abstract

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.