Suppose that $X,Y$ are positive random variables such that $P(Y \le \tau)\ge 1-\epsilon$ where $\epsilon$ is small, i.e. $Y$ is bounded with high probability.
Can bounds be constructed for $E[XY]$ that depend on $E[X], \tau,$ and $\epsilon$?
The closest result I have seen is Theorem 15 of this pdf which is a bound on $E[Y]$. What about for the product?
Can probabilistic statements also be made between $E[XY]$ and X?
Given that you are not assuming independence, integrability, positivity or anything else you can not. The result you linked relies on the fact that the inequality you gave holds for every $\tau$ i.e. $P(Y \leq \tau) \geq 1-f(\tau)$ for some function f.