Upper bounds for E(f(x))

Question

Given a discrete random variable X, with finite mean $\mu$ and variance $\sigma^2$ and a convex function f(x), what is the tighter upper bound one can give for E(f(x))? I am looking for something in this line: Lower-bound for $E(\log{((x+1)!)}$)? . Maybe that's the best, I'm just curious. Does the result have a well-known name or a paper associated?