A complement of the Ando-Hiai inequality

摘要

In this paper, we present a complement of a generalized Ando-Hiai inequality due to Fujii and Kamei [M. Fujii, E. Kamei, Ando-Hiai inequality and Furuta inequality, Linear Algebra Appl. 416 (2006) 541-545]. Let A and B be positive operators on a Hilbert space H such that 0 < m(1) <= A <= M-1 and 0 < m(2) <= B <= M-2 for some scalars m(i) <= M-i (i = 1,2), and alpha epsilon [0, 1]. Put h(i) = M-i/m(i) for i =1, 2. Then for each 0 < r <= 1 and s >= 1 [GRAPHICS] where A#(alpha) B := A(1/2)(A(-1/2) BA(-1/2))(alpha) A(1/2) is the alpha-geometric mean and a generalized Kantorovich constant K (h, p) is defined for It > 0 as K (h,p) := h(p)-h/(p-1)(h-1) (p-1/p h(p)-1/h(p)-h)(p) for all real numbers p epsilon R. (C) 2008 Elsevier Inc. All rights reserved.