Let me try to make the problem more "intriguing". The inequality does not hold in general (it is very easy to build counterexamples). BUT it always holds for independent variables. Here is a sketch of a proof:
For any variables and it holds that
Now consider that, for any constant :
By integrating with respect to the distribution of , and using independence, I get to the point.
(A rigorous proof is somewhat longer, but at least you see the way to get there)
Now can we prove my inequality for some families of correlated variables?
With the normal we can (a proof is coming right away)