Log-majorizations of Lemos–Soares type for the metric geometric and spectral geometric means

Abstract

The metric geometric and spectral geometric means are two essential concepts in the theory of matrix mean. In [15], Lemos and Soares asked whether there exists the following log-majorization relation for \(A, B\ge 0\), \(t\in [0,1]\),

$$\begin{aligned} s(A^t(A\sharp _{t}B)B^{1-t}) {\mathop \prec \limits _{(\log )}} s(AB). \end{aligned}$$

Recently, Gan and Kim prove alternative versions of above log-majorization for the metric geometric and spectral geometric means in [8]. In this paper, by several Furuta-type inequalities, we obtain several log-majorizations of Lemos–Soares type for the metric geometric mean and weak log-majorizations of Lemos–Soares type for the spectral geometric mean, which extend the recent results from Gan and Kim and other related results.