Refinements of a reversed AM-GM operator inequality

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 $\alpha=\max\left\{\frac{(M+m)^2}{4Mm},\frac{(M+m)^2}{4^\frac{2}{p}Mm}\right\}$.