Given a positive function f on (0, ∞) and a non-zero real parameter θ, we consider a function I~θ _f(A,B,X) = tr X~*(f(L_AR~(-1) _B)R_B)~(-θ)(X)in three matrices A, B > 0 and X. This generalizes the notion of monotone metrics on positive definite matrices, and in the literature θ = ±1 has been typical. We investigate how operator monotony of f is sufficient and/or necessary for joint convexity/concavity of I~θ _f(A,B,X). Similar discussions are given for quasi-entropies and quantum skew informations.
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