Generalized Pólya–Szegö type inequalities for some non-commutative geometric means

In this paper, we shall show generalized Pólya–Szegö 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.