Let $X$ and $Y$ be two positive continuous random variable. I have noticed that
$$cov(\sqrt{X}, \sqrt{Y}) \leq cov(X,Y) \leq cov(Y^2, X^2).$$
I am wondering if there is a general rule that say that if (for example) $E[f(X)] < E[X]$ and $E[f(Y)] < E[Y]$ for some measurable function f defined on $[0, \infty)$, then we may be able to say that
$$cov(f(X), f(Y)) \leq cov(X,Y)?$$