Nonnegativity bound for a prolate logarithmic-potential distribution ($q^2\le (16\sqrt{2}-e)/(16\sqrt{2}-2e)$)

For the logarithmic-potential two-integral distribution, $q^2>1$ makes $A<0$. Accessible phase space satisfies $L_z^2{e}^{-2E/V^2}\le V^2/e$, attained by equatorial circular motion. Hence the galactic distribution function is nonnegative precisely when $B+A{V}^2/e\ge 0$, giving the displayed upper bound. A larger prolate flattening parameter produces negative phase-space density even though the spatial mass density remains positive.