I'll be using the symbols from my writeup on the Jensen inequality, so you'll want to read that first. These proof of ... writeups all start the same, don't they? I get so bored, talking to myself and nobody ever listens. I mean, I could just write EDB borg xy=17 probability and people would still softlink it to insulting titles but still vote it up and C! it.
OK, so let's write x = EX. Being convex, the function f has a supporting line at x. That is, there exists a linear function L(t) = at+b for which L(x)=f(x) and L(t) ≤ f(t) for all t ("f remains above L").
The simplicity of the truth of the inequality is revealed!
Ef(X) ≥ EL(X) =
b + aEX =
b + ax = L(x) = f(x) = f(EX)
Q.E.D.