A generalization of Araki's log-majorization

We generalize Araki's log-majorization to the log-convexity theorem for the eigenvalues of $Φ(A^p)^{1/2}Ψ(B^p)Φ(A^p)^{1/2}$ as a function of $p\ge0$, where $A,B$ are positive semidefinite matrices and $Φ,Ψ$ are positive linear maps between matrix algebras. A similar generalization of the log-majorization of Ando-Hiai type is given as well.

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