New Inequalities of the Kantorovich Type With Two Negative Parameters

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.

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