An S-Shaped Buckley–Leverett Family Outside Rastegaev's Logarithmic-Curvature Class

The fractional flow function in the Buckley--Leverett equation is conventionally assumed to be S-shaped. Rastegaev recently established a sufficient condition for this property based on monotonicity of \(m''/m'\), and showed that strict convexity of the phase mobilities alone is not sufficient. This note demonstrates that Rastegaev's criterion is not necessary, by exhibiting an explicit one-parameter polynomial family of strictly convex mobility functions, \[ m_\alpha(s)=s^2(1+\alpha s^4)=s^2+\alpha s^6,\qquad \alpha\ge 0, \] for which the symmetric fractional flow function \[ f_\alpha(s)=\frac{m_\alpha(s)}{m_\alpha(s)+m_\alpha(1-s)} \] retains its S-shape for every \(\alpha\ge 0\): \(f_\alpha''>0\) on \((0,1/2)\), \(f_\alpha''(1/2)=0\), and \(f_\alpha''<0\) on \((1/2,1)\). The mobility lies outside Rastegaev's class once \(\alpha>(5-2\sqrt 5)/15\approx 0.0352\). The proof reduces the sign of \(f_\alpha''\) to four explicit polynomial inequalities on \([0,1/4]\), certified by the convex-hull property of Bernstein coefficients. The same Bernstein certificate applies, without modification, to \(m_\alpha(s)=s^2(1+\alpha s^q)\) for every integer \(q\in\{2,3,4,5,6\}\); at \(q=7\) the certificate just fails.