Quantum information functionals and their algebraic properties play an important role in quantum information theory. Recent developments in the theory of operator monotone and operator convex functions and related topics have simplified earlier results and also led to new insights. One example is the convexity of chi-square divergences. We discuss an intriguing strong order relation for measures of quantum information exemplified by the Wigner-Yanase-Dyson skew information measures and other WYD-like skew information measures. We finally introduce and discuss the notion of classical mixing relative to a preferred quantum statistics.