New Inequalities of the Kantorovich Type With Two Negative Parameters

TL;DR

This paper establishes new inequalities of the Kantorovich type involving two negative parameters for positive operators, extending existing bounds and introducing inequalities for the chaotic order.

Contribution

It introduces novel Kantorovich inequalities with two negative parameters for positive operators, expanding the scope of operator inequalities.

Findings

Derived bounds for positive operators with two negative parameters.

Established inequalities involving the chaotic order.

Extended classical inequalities to new parameter ranges.

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.