How general is the global density slope-anisotropy inequality?
  Following the seminal result of An & Evans, known as the central density
slope-anisotropy theorem, successive investigations unexpectedly revealed that
the density slope-anisotropy inequality holds not only at the center, but at
all radii in a very large class of spherical systems whenever the phase-space
distribution function is positive. In this paper we derive a criterion that
holds for all spherical systems in which the augmented density is a separable
function of radius and potential: this new finding allows to unify all the
previous results in a very elegant way, and opens the way for more general
investigations. As a first application, we prove that the global density
slope-anisotropy inequality is also satisfied by all the explored additional
families of multi-component stellar systems. The present results, and the
absence of known counter-examples, lead us to conjecture that the global
density slope-anisotropy inequality could actually be a universal property of
spherical systems with positive distribution function.
