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, measurable, and increasing function.

My question is: are $f(X)$ and $f(Y)$ still positively correlated?

Thanks a lot.