Convexity of quasi-entropy type functions: Lieb's and Ando's convexity theorems revisited

Given a positive function $f$ on $(0,\infty)$ and a non-zero real parameter $\theta$, we consider a function $I_f^\theta(A,B,X)=Tr X^*(f(L_AR_B^{-1})R_B)^\theta(X)$ in three matrices $A,B>0$ and $X$. In the literature $\theta=\pm1$ has been typical. The concept unifies various quantum information quantities such as quasi-entropy, monotone metrics, etc. We characterize joint convexity/concavity and monotonicity properties of the function $I_f^\theta$, thus unifying some known results for various quantum quantities.