A generalization of Araki's log-majorization

We generalize Araki's log-majorization to the log-convexity theorem for the eigenvalues of Φ(Ap)1/2Ψ(Bp)Φ(Ap)1/2Φ(Ap)1/2Ψ(Bp)Φ(Ap)1/2 as a function of p≥0p≥0, where A, B   are positive semidefinite matrices and Φ, Ψ are positive linear maps between matrix algebras. Similar generalizations of the log-majorization of Ando–Hiai type for the weighted geometric mean A#αB are also given, including a lemma on general operator means AσE of A and a projection E.