= Concave Legendre dual
{title2=$g(z)=\inf_s(zs-f(s))$}
For a differentiable strictly <concave function> $f$ whose derivative is a bijection of $\mathbb R$, define
$$
g(z)=\inf_{s\in\mathbb R}\{zs-f(s)\}=zh(z)-f(h(z)),\qquad h=(f')^{-1}.
$$
The unique minimizing point is $h(z)$, and $g'=h$ follows by comparing minimizers at adjacent arguments, even when $h$ is not differentiable. This dual is concave and equals $-(-f)^*(-z)$ in terms of the <convex conjugate>. If $f''<0$, then $g''=1/f''(h)$. For $f(s)=cs-s^2/2$, $g(z)=-(z-c)^2/2$.
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