Central-density upper bound for a radial stellar frequency (${\omega }_{\min }^2\le (3\gamma -4)4\pi G{\rho }_c/3$)

For constant adiabatic exponent $\gamma >4/3$ and a star whose central mass density is its maximum, the homologous radial fluid displacement ${\xi }_r=r$ makes the derivative term in the weighted stellar pulsation Rayleigh quotient vanish. The remaining quotient is $(3\gamma -4)$ times a positive weighted average of $g/r$. Since $m(r)\le 4\pi {\rho }_c{r}^3/3$, $g/r\le 4\pi G{\rho }_c/3$, proving the bound. For $\gamma <4/3$, the same trial quotient is negative and proves radial instability; the inequality with the central density must not be extended with its sign unchanged.