An H-theorem for violent relaxation?

Recently, Tremaine, Hénon & Lynden-Bell have purported to prove a ‘coarse-grained’ H-theorem inequality relevant to the phenomenon of violent relaxation in collisionless systems of stars. Specifically, they claim that any convex function of a suitably coarse-grained distribution function defines an ‘H-function . . . which is non-decreasing with time.’ It is indicated here that what they have actually shown, albeit extremely interesting, is a somewhat weaker result which does not in fact constitute a true H-theorem inequality.