On a reverse of Ando--Hiai inequality - Seo, Yuki (Banach journal of mathematical analysis, 2010)
In this paper, we show a complement of Ando-Hiai inequality: Let A and B be positive invertible operators on a Hilbert space H and [alpha] [member of] [0,1]. If A [??] [alpha] [less than or equal to] B [less than or equal to] I, then [A.sup.r] [[??].sup.[alpha]] Br [less than or equal to] [[parallel](A [[??].sub.[alpha]] [B)].sup.-1] [parallel].sup.1-r] I for all 0 < r [less than or equal to] 1, where I is the identity operator and the symbol [parallel] x [parallel] stands for the operator norm. 2000 Mathematics Subject Classification. Primary 47A63; Secondary 47A30, 47A64. Key words and phrases. Ando-Hiai inequaqlity, positive operator, geometric mean.