Epicyclic frequency bounds for monotone spherical density ($\Omega \le \kappa \le 2\Omega$)

For a nonnegative, radially nonincreasing spherical mass density, ${\Omega }^2=GM(r)/r^3$ and ${\kappa }^2={\Omega }^2+4\pi G\rho (r)$. Since $\rho (r)$ does not exceed the mean enclosed density, the radial epicyclic frequency lies between $\Omega$ and $2\Omega$. General axisymmetric disks need not satisfy this bound.