I am looking for good upper bounds on $E[(f(X)-g(X))^2]$. For example, here are two bound that I derived: $$ E[(f(X)-g(X))^2] \le E[4 \max(f(X)^2, g(X)^2)] $$ where I used $(x-y)^2 \le 4 \max(x^2,y^2)$. Unfortunatly, this bounds is still hard to compute since it is not true that $ E[4 \max(f(X)^2, g(X)^2)] \le 4 \max(E[f(X)^2, E[g(X)^2])$ . At least I don't think so.
Another, bound uses $(x-y)^2 \le 2x^2+2y^2$ then $$ E[(f(X)-g(X))^2] \le 2E[f(X)^2]+2E[g(X)^2] $$
I was wondering if there are batter bounds that are easy to compute?
You have the Cauchy-Schwarz inequality, $$ \lvert E[f(X) g(X)] \rvert \leq \sqrt{E[f(X)^2] E[g(X)^2]}. $$ This then gives you, after expanding out the brackets and refactoring, $$ E[(f(X)-g(X))^2] \leqslant \left( (E[f(X)^2])^{1/2} + (E[g(X)^2])^{1/2} \right)^2 $$