Convex function and expectation

Question

I was wondering: if f is a convex function and X a random variable, what does E(f(X)) = f(E(X)) implies?

Thanks a lot, David

Answer 1

If convex, the equality is $\geq$. If concave, the equality is $\leq$. If both a true, the function is both concave and convex. Therefore, it is linear (try showing this using definitions of convex, concave and linear). Also, I'm not sure, but I think such implication holds without the assumption that f is convex.

Answer 2

It implies that $x \mapsto f$ is a linear function.