Log-majorizations between quasi-geometric type means for matrices

Abstract: In this paper, for $\alpha\in(0,\infty)\setminus{1}$, $p&gt;0$ and positive semidefinite matrices $A$ and $B$, we consider the quasi-extension $$\mathcal{M}<em>{\alpha,p}(A,B):=\mathcal{M}</em>\alpha(A<sup>p,B<sup>p)<sup>{1/p}$$ of several $\alpha$-weighted geometric type matrix means $\mathcal{M}<em>\alpha(A,B)$ such as the $\alpha$-weighted geometric mean in Kubo--Ando's sense, the R\'enyi mean, etc. The log-majorization $$\mathcal{M}</em>{\alpha,p}(A,B)\prec_{\log}\mathcal{N}<em>{\alpha,q}(A,B)$$ is examined for pairs $(\mathcal{M},\mathcal{N})$ of those $\alpha$-weighted geometric type means. The joint concavity/convexity of the trace functions $\mathrm{Tr}\,\mathcal{M}</em>{\alpha,p}$ is also discussed based on theory of quantum divergences.