Abstract

We discuss two variables version of the Ando-Hiai inequality: For A, B > 0 and α ∈ [0, 1], if A Open image in new window B ≤ I, then Open image in new window Here Open image in new window is the α-geometric mean in the sense of Kubo-Ando. In this context, the Furuta inequality is understood as the one-sided version (the case of s = 1): If A Open image in new window B ≤ I, then Open image in new window As a consequence, the Furuta inequality has an alternative simple proof. In addition, we point out that the obtained inequality is understood as the case t = 1 in the grand Furuta inequality.

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