Inequality for an expectation involving a product of densities

Question

Let $X$ and $Y$ be two real (independent) random variables with pdfs $f$ and $g$ respectively, and such that their expectations exist and are finite and $\mathbb{E}[X] \leq \mathbb{E}[Y]$. Define a new pdf as $$h(z) = \frac{f(z) g(z)}{\int f(z) g(z) \; d z}$$ Let $Z$ be a real random variable with pdf $h$ (also independent of $X$ and $Y$). Assume $\mathbb{E}[Z]$ exists and is finite. I wonder what are sufficient and necessary conditions for the following to hold $$ \mathbb{E}[X] \leq \mathbb{E}[Z] \leq \mathbb{E}[Y] $$ Thanks