an:05702400 Zbl 1186.47014 Seo, Yuki On a reverse of Ando-Hiai inequality. EN Banach J. Math. Anal. 4, No. 1, 87-91, electronic only (2010). ISSN 1735-8787/e 2010

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*47A63 47A30 47A64 Ando-Hiai inequality; positive operator; geometric mean Summary: We show a complement of the Ando-Hiai inequality: Let $A$ and $B$ be positive invertible operators on a Hilbert space $H$ and $\alpha\in[0,1]$. If $A\sharp_\alpha B\le I$, then? $$A^r\sharp_\alpha B^r\le \|(A\sharp_\alpha B)^{-1}\|^{1-r}I \quad\text{for all }0<r\le 1,$$? where $I$ is the identity operator and $\|\cdot\|$ stands for the operator norm.