A satellite of the grand Furuta inequality and its application

The grand Furuta inequality has the following satellite (SGF:t is an element of [0, 1]), given as a mean theoretic expression: A >= B > 0, t is an element of [0, 1] double right arrow A(-r+t)#(1-t+t/(p-t)s+r)(A(t)#sB(p)) <= B (p t)s+r for r >= t; p,s >= 1, where #(alpha), is the alpha-geometric mean and Lis (s is not an element of [0, 1]) is a formal extension of It,. It is shown that (SGF; t is an element of [0, 1]) has the Lowner-Heinz property, i.e. (SGF; t = 1) implies (SGF;t) for every t is an element of [0, 1]. Furthermore, we show that a recent further extension of (GFI) by Furuta himself has also the Lowner-Heinz property. (C) 2011 Elsevier Inc. All rights reserved.