If $X,Y$ are positively correlated, are $f(X),f(Y)$ also positively correlated for a positive increasing $f$?

Question

Suppose that $X$ and $Y$ are positive and square-integrable random variables, such that $X$ and $Y$ are positively correlated, i.e., $\mathbb{E}[XY] - \mathbb{E}[X]\mathbb{E}[Y] \geq 0$. Let $f: \mathbb{R} \to \mathbb{R}_+$ be a positive and increasing function. My question is: Are $f(X)$ and $f(Y)$ positively correlated? Thanks very much.

Answer 1

Let $X\sim U(0,2)$, $Y=(X-1)^2$, and $f(x)=(\sqrt{x}+1)1\{x\ge 0\}+e^x1\{x<0\}$. Then $f$ is a positive, increasing function and $\mathrm{Cov}(X,Y)=0$. However, $$ \mathrm{Cov}(f(X),f(Y))=\left(\frac{4}{15}-\frac{\sqrt{2}}{5}\right)<0. $$