Given the function $g(\vec{x})=\log \sum_i \exp(x_i)$, I am curious which functions $f$ satisfy $f(g(\vec{x})) \geq g(\vec{f(x)})$?
Let's let $x \in \mathbb{R}^N$ to be concrete.
Useful properties of $g$: convex, monotone in each argument. For instance, I can show $f(x)=\alpha x$ satisfies this inequality for $\alpha \geq 1$ (and the reverse inequality for $0< \alpha < 1$) by Hölder's inequality.
How can I generally find other functions which satisfy this (or the reversed) inequality? What if we also assume $f$ to be monotonic, convex/concave etc.?
As another possible function that may work: a (continuous) piecewise set of linear functions with increasing slope (implies convexity). I am having a lot of trouble proving this though because of the different subcases that can arise (based on the different branches of the function).