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.

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 $H(x,y)$ is $\min(F(x),G(y))$. By Hoeffding's Lemma we get that $$E(XY)\leq E(X)E(Y)+\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\big[\min(F(x),G(y))-F(x)G(y)\big]dxdy$$