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I would try yo apply Hoeffding's Lemma, who used his result to identify the bivariate cdfs with given marginal cdfs that minimize or maximize correlation. Let (X,Y) be a random vector with bivariate cdf H, let F and G be their marginal cdfs, respectively. It is well known that a sharp upper bound for H is min(F,G). By Hoeffding's Lemma we get that E(XY) is bounded above by E(X)E(Y) + DI[min(F,G) - F*G], where DI stands for the double integral on the whole plane. But I don't know if this is a sharp bound or if at least it is better than the one you may obtain by Hölder's inequality.