Wow, this took me a surprisingly long time to show. Yes, it is true, provided that you allow the inequality to be attained (if X is a point mass you get a covariance of 0).
Let . Let and . First, note that is convex, and hence by Jensen's Inequality we have .
Next, note that it suffices to show . By definition this is the statement that
and solving for this is the statement , which is always true by Jensen's Inequality.