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We introduce a generalization of relative entropy derived from theWigner-Yanase-Dyson entropy and give a simple, self-contained proof that it isconvex. Moreover, special cases yield the joint convexity of relative entropy,and for the map (A,B) --> Tr K^* A^p K B^{1-p} Lieb's joint concavity for 0 < p< 1 and Ando's joint convexity for 1 < p < 2. This approach allows us to obtainconditions for equality in these cases, as well as conditions for equality in anumber of inequalities which follow from them. These include the monotonicityunder partial traces, and some Minkowski type matrix inequalities proved byLieb and Carlen for mixed (p,q) norms. In all cases the equality conditions areindependent of p; for extensions to three spaces they are identical to theconditions for equality in the strong subadditivity of relative entropy.

A Unified Treatment of Convexity of Relative Entropy and Related Trace Functions, with Conditions for Equality