  For a wide class of self-gravitating systems, we show that if the density is
cusped like 1/r^{gamma} near the center, then the limiting value of the
anisotropy parameter beta = 1 - <v_T^2>/(2<v_r^2>) at the center may not be
greater than (gamma/2). Here, <v_r^2> and <v_T^2> are the radial and tangential
velocity second moments. This follows from the non-negativity of the phase
space density. We compare this theorem to other proposed relations between the
cusp slope and the central anisotropy to clarify their applicabilities and
underlying assumptions. The extension of this theorem to tracer populations in
an externally imposed potential is also derived. In particular, for stars
moving in the vicinity of a central black hole, this reduces to gamma >=
beta+(1/2), indicating that an isotropic system in Keplerian potential should
be cusped at least as steep as 1/r^{0.5}. Similar limits have been noticed
before for specific forms of the distribution function, but here we establish
this as a general result.
