 Uniformly convex scalar flux

In the common PDE convention, a uniformly convex scalar flux is a $C^2$ flux satisfying $F^{\prime \prime }(z)\ge \kappa >0$ throughout the specified state interval, with one constant $\kappa$. Equivalently, $F(z)-\kappa z^2/2$ is a convex function. This guarantees strict inequality below nontrivial chords and strict ordering of the two endpoint derivatives around the chord slope. On the whole real line it is incompatible with a globally bounded first derivative.