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 M⏐, ₏ (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 M⏐, ₏ (A, B) _N⏐, ₐ (A, B) is examined for pairs (M, N) of those α-weighted geometric type means. The joint concavity/convexity of the trace functions Tr\, M⏐, ₏ is also discussed based on theory of quantum divergences.