For a non-convex function f, how to find a function g such that $g\circ f$ is strictly convex?

Question

The following function $f(x)={1\over (1+e^{-x})}$ is non-convex but $\ln(f(x))$ is convex. Given a non-convex function $f$, can we find a function $g$ such that $g\circ f$ is strictly convex? If yes, how to find such a g? If not, what property should f meet when such a g can be found?

Answer 1

Таке any convex function, say $s(x)=\exp(x)$. Let $f(x)$ be monotone. Solve $g(f(x))=s(x)=\exp(x), f(x)=t, x=p(t)$ and so $g(t)$=s(p(t))=\exp(p(t))$ will be OK!