I have two random variables $X \in [0,1], Y \in [a,1]$, and I would like to lower-bound the expectation $E[X Y]$, preferably in terms of $E[X]$ and $E[Y]$. I would accept anything better than the trivial $E[X Y] \geq a E[X]$.
In general, the worst possible distribution would be if $X, Y$ were anti-correlated, that is, when $Y$ is large then $X$ is small. However, I also have bounds on these namely that with certainty $$ X \geq Y^k $$ for some constant $k \geq 1$. Does this give any way to lower-bound $E[X Y]$?