Any bound on the Jensen's inequality with absolute value?

Question

So we have the jensen's inequality: $$|EX| \leq E|X|$$

Any bound on the Jensen gap (upper bound or lower bound)? $$\text{gap}=E|X| - |EX|$$

Answer 1

The gap can be arbitrarily large. For instance, if $X$ is a random variable so that $X(0) = -N$ and $X(1)=N$, and the events $0$ and $1$ have probability $1/2$, then $|E(X)| = |\frac{1}{2}N - \frac{1}{2}N|=0$, but $E(|X|) = N$.

Answer 2

$0 \le |EX| \leq E|X|$ so $$\text{gap} \le E|X|.$$ Reijo provided an example where $\text{gap} = E|X|$.