On the universality of the global slope -- anisotropy inequality

Recently, some intriguing results have lead to speculations whether the central density slope -- velocity dispersion anisotropy inequality (An & Evans) actually holds at all radii for spherical dynamical systems. We extend these studies by providing a complete analysis of the global slope -- anisotropy inequality for all spherical systems in which the augmented density is a separable function of radius and potential. We prove that these systems indeed satisfy the global inequality if their central anisotropy is $\beta_0\leq 1/2$. Furthermore, we present several systems with $\beta_0 > 1/2$ for which the inequality does not hold, thus demonstrating that the global density slope -- anisotropy inequality is not a universal property. This analysis is a significant step towards an understanding of the relation for general spherical systems.