Generalized Pólya-Szegő type inequalities for some non-commutative geometric means (Q1940323)

For any integer \(n \geq 3\), there are two important definitions of geometric means of \(n\) positive invertible operators: One is the higher order weighted geometric mean due to \textit{J. Lawson} and \textit{Y. Lim} [Colloq. Math. 113, No. 2, 191--221 (2008; Zbl 1160.47016)], which is an extension of the geometric mean due to \textit{T. Ando}, \textit{C.-K. Li} and \textit{R. Mathias} [Linear Algebra Appl. 385, 305--334 (2004; Zbl 1063.47013)]. The other is the weighted chaotic geometric mean; see \textit{R. D. Nussbaum} and \textit{J. E. Cohen} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 15, No. 2, 239--308 (1988; Zbl 0756.47018)]. From the viewpoint of generalized Pólya-Szegő type inequalities, which are ratio type reverses of the Cauchy-Schwarz inequality, for \(n\) positive invertible operators, the author of present paper compares these two geometric means. In this investigation, he uses the Specht ratio, which is the upper bound of a ratio type reverse of the weighted arithmetic-geometric mean inequality. For other noncommutative versions of the Pólya-Szegő inequality, one may consult \textit{M. Fujii} et al. [Nihonkai Math. J. 8, No. 2, 117--122 (1997; Zbl 0997.47505)] as well as \textit{M. S. Moslehian} and \textit{L.-E. Persson} [Math. Inequal. Appl. 12, No. 4, 701--709 (2009; Zbl 1188.46037)].