  We study here the elementary properties of the relative entropy
$\cH(A,B)=\tr[\phi(A)-\phi(B)-\phi'(B)(A-B)]$ for $\phi$ a convex function and
$A,B$ bounded self-adjoint operators. In particular, we prove that this
relative entropy is monotone if and only if $\phi'$ is operator monotone. We
use this to appropriately define $\cH(A,B)$ in infinite dimension.
