Log-majorizations between quasi-geometric type means for matrices

In this paper, for $\alpha \in (0,\mathrm{\infty })\setminus \{1\}$, $p>0$ and positive semidefinite matrices $A$ and $B$, we consider the quasi-extension ${\mathcal{M}}_{\alpha ,p}(A,B):={\mathcal{M}}_{\alpha }(A^p,B^p{)}^{1\mathrm{/}p}$ of several $\alpha$-weighted geometric type matrix means ${\mathcal{M}}_{\alpha }(A,B)$ such as the $\alpha$-weighted geometric mean in Kubo–Ando’s sense, the Rényi mean, etc. The log-majorization ${\mathcal{M}}_{\alpha ,p}(A,B){\prec }_{\log }{\mathcal{N}}_{\alpha ,q}(A,B)$ is examined for pairs $(\mathcal{M},\mathcal{N})$ of those $\alpha$-weighted geometric type means. The joint concavity/convexity of the trace functions $\mathrm{T}\mathrm{r}{\mathcal{M}}_{\alpha ,p}$ is also discussed based on theory of quantum divergences.