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.
