Suppose we have two random variables $X$ and $Y$ where $\mathrm{cov}(X,Y) = a > 0$. Let $U(X)$ be an increasing and concave function. I am interested in whether we ever can say something about the covariance between the concave transformation $U(X)$ and $Y$.
More precisely, are there any conditions under which we can sign $\mathrm{cov}(U(X),Y)$ or even say $\mathrm{cov}(U(X),Y) > \mathrm{cov}(X,Y)$?
I suspect no in the general case, but what if $U(X)$ has the following form (iso-elastic utility in economics): $$U(X;\eta) = \frac{X^{1-\eta}}{1-\eta}$$ where $\eta \geq 0, \eta \neq 1$.
One strategy I was thinking may work in the special case was to differentiate the covariance with respect to $\eta$ since $U(X;0) = X$ and larger $\eta$ increases the concavity of the transformation.