Generalized Polya-Szego type inequalities for some non-commutative geometric means

In this paper, we shall show generalized Polya-Szego type inequalities of n positive invertible operators on a Hilbert space for any integer n >= 3 in terms of the following two typical non-commutative geometric means, that is, one is the higher order weighted geometric mean due to Lawson-Lim which is an extension of the Ando-Li-Mathias geometric mean, and the other is the weighted chaotic geometric mean. Among others, the Specht ratio plays an important role in our discussion, which is the upper bound of a ratio type reverse of the weighted arithmetic-geometric mean inequality. (C) 2011 Elsevier Inc. All rights reserved.