  We introduce a generalization of relative entropy derived from the
Wigner-Yanase-Dyson entropy and give a simple, self-contained proof that it is
convex. 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 obtain
conditions for equality in these cases, as well as conditions for equality in a
number of inequalities which follow from them. These include the monotonicity
under partial traces, and some Minkowski type matrix inequalities proved by
Lieb and Carlen for mixed (p,q) norms. In all cases the equality conditions are
independent of p; for extensions to three spaces they are identical to the
conditions for equality in the strong subadditivity of relative entropy.
