= Central-density upper bound for a radial stellar frequency
{title2=$\omega_{\min}^2\leq(3\gamma-4)4\pi G\rho_c/3$}
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 $m(r)\leq4\pi\rho_cr^3/3$, $g/r\leq4\pi G\rho_c/3$, 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.
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