Let $X,Y$ be two (dependent) random variables with $\mathbb{P}(X\ge 0)=\mathbb{P}(Y\ge 0)=1$.

I want to find a tight lower bound of $\mathbb{E}(XY)$ when $X,Y$ are non-negative, almost surely.

Note that one trivial lower bound is $0$. But I want to know if there is an elegant method in literature for finding a tight lower bound, which has information of only the individual moments of the random variables $X,Y$.

I realize that the following lower bound can be obtained simply by using Cauchy-Scwartz inequality $$\mathbb{E}(XY)\ge \mu_X\mu_Y-\sigma_X\sigma_Y,$$ where $\mu_X=\mathbb{E}X,\ \sigma_X^2=\mathrm{Var}(X)$, and similarly $\mu_Y,\sigma_Y$ defined for $Y$. However, I don't know that if $X,Y$ are non-negative, almost surely, then whether this lower bound is non-negative. A little bit of algebra shows that this claim is equivalent to the following claim $$\frac{\mu_X^2}{\mu_{X^2}}+\frac{\mu_Y^2}{\mu_{Y^2}}\ge 1.$$ However, when $X=Y$, this claim is equivalent to proving that $$\mu_X^2\ge \frac{\mu_{X^2}}{2},$$ which, however, is false in general because Holder's inequality implies that $|\mu_X|\le \frac{\sqrt{\mu_{X^2}}}{\sqrt{2}}$. Any ideas how I can proceed? Thanks in advance.