Refinements of a reversed AM-GM operator inequality

TL;DR

This paper presents refined inequalities related to the reverse AM-GM operator inequality, providing tighter bounds for positive operators on a Hilbert space using linear maps and operator means.

Contribution

It introduces new refined inequalities for the reverse AM-GM operator inequality, extending previous results with explicit bounds involving positive operators and linear maps.

Findings

Derived a new operator inequality with explicit bounds.

Extended the reverse AM-GM inequality to a broader class of operators.

Provided conditions for the inequality involving positive operators and linear maps.

Abstract

We prove some refinements of a reverse AM-GM operator inequality due to M. Lin [Studia Math. 2013;215:187-194]. In particular, we show the operator inequality \begin{eqnarray*} \Phi^p\left(A\nabla_\nu B+2rMm(A^{-1}\nabla B^{-1}-A^{-1}\sharp B^{-1})\right)\leq\alpha^p\Phi^p\left(A\sharp_\nu B\right), \end{eqnarray*} where $A,B$ are positive operators on a Hilbert space such that $0<m \leq A, B \leq M$ for some positive numbers $m, M$, $\Phi$ is a positive unital linear map, $\nu\in[0,1]$, $r=\min\{\nu,1-\nu\}$, $p>0$ and $\alpha=\max\left\{\frac{(M+m)^2}{4Mm},\frac{(M+m)^2}{4^\frac{2}{p}Mm}\right\}$.