If the world is nice, let $f$ be a positive and strictly increasing differentiable function, i.e. $f>0$ and $f'>0$.
Is the following statement true?
There always exist $p\in\mathbb{R}$, such that $g = f^p$ is convex.
How about we replace 'convex' with 'concave'?
I came across this question in my study of submodularity, a notion of concavity in combinatorial optimization. If you are familiar with submodularity, is it true that for any monotonic function $f$, I can take some power of $f$ and make it submodular?