A self-improvement to the Cauchy–Schwarz inequality

We present a self improvement to the Cauchy–Schwarz inequality, which in the probability case yields [E(XY)]2≤E(X2)E(Y2)−(|E(X)|Var(Y)−|E(Y)|Var(X))2. It is to be noted that the additional term to the inequality only involves the marginal first two moments for XX and YY, and not any joint property. We also provide the discrete improvement to the inequality.