 Concave Legendre dual

For a differentiable strictly concave function $f$ whose derivative is a bijection of $\mathbb{R}$, defineThe unique minimizing point is $h(z)$, and $g^{\prime }=h$ follows by comparing minimizers at adjacent arguments, even when $h$ is not differentiable. This dual is concave and equals $-(-f{)}^{\ast }(-z)$ in terms of the convex conjugate. If $f^{\prime \prime }<0$, then $g^{\prime \prime }=1/f^{\prime \prime }(h)$. For $f(s)=cs-s^2/2$, $g(z)=-(z-c{)}^2/2$.