Log-majorizations between quasi-geometric type means for matrices

In this paper, for α∈(0,∞)\1\, $p>0$ and positive semidefinite matrices A and B, we consider the quasi-extension Mα,p(A,B):=M_α(Aᵖ,Bᵖ)1/p of several $α$-weighted geometric type matrix means M_α(A,B) such as the $α$-weighted geometric mean in Kubo--Ando's sense, the Rényi mean, etc. The log-majorization Mα,p(A,B)logNα,q(A,B) is examined for pairs (M,N) of those $α$-weighted geometric type means. The joint concavity/convexity of the trace functions Tr\,Mα,p is also discussed based on theory of quantum divergences.