From $\mathbb{Cov}(X^2,Y)$ to $\mathbb{Cov}(X,Y)$

Question

Say I have the expression for $\mathrm{Cov}(X^2,Y)>0$, where both $X$ and $Y$ are known non-negative random variables. What if anything can we say about $\mathrm{Cov}(X,Y)$ without having to derive it from scratch? I think we can say that $\mathrm{Cov}(X,Y)>0$, but can we say anything more, given the expression for $\mathrm{Cov}(X^2,Y)$?

Answer 1

I think it can say very poor . For example if you take $Y=X^2$, then $Cov(X^2,Y)=1$. But if you take positive values for X then $Cov(X,Y)>0$, and for negative values for X, $Cov(X,Y)<0$. Posibly if you take any type of values, $Cov(X,Y)\cong 0$.