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*} Φ^p\left(A\nabla_νB+2rMm(A^{-1}\nabla B^{-1}-A^{-1}\sharp B^{-1})\right)\leqα^pΦ^p\left(A\sharp_νB\right), \end{eqnarray*} where $A,B$ are positive operators on a Hilbert space such that $0<m\le A,B\le M$ 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 $\alpha =\max \left\{\frac{(M+m{)}^2}{4Mm},\frac{(M+m{)}^2}{4^{\frac{2}{p}}Mm}\right\}$.

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