Jensen's inequality, $\mathbb{E}[\phi(X)]\geq\phi(\mathbb{E}[X])$, holds for any convex function $\phi$ and random variable $X$.
Given that $\mathbb{E}[\phi(X)]=\phi(\mathbb{E}[X])$ for some non-degenerate random variable $X$, what can we say about $\phi$? any why? (assuming that it is not necessarily convex). What about the case in which $\phi$ is differentiable?
I wonder if there is an interesting universal property on all function $\phi$ that lead to such equality.
Best Answer
I don't think there is much that can be said without more restrictions. If $X$ is some random variable such that $P(X=E[X]) = 0$, then for any function $\phi$ you can define $\tilde{\phi}(t) = \begin{cases} \phi(t) & t \ne E[X] \\ E[\phi(X)] & t = E[X] \end{cases}$ to get $\tilde{\phi}(E[X]) := E[\phi(X)] = E[\tilde{\phi}(X)]$.