Suppose $f(x,\mu)$ and $g(x,\mu)$ are both positive functions, and for any $x$, $\frac{f(x,\mu)}{g(x,\mu)}$ increases with $\mu$.
Let $X$ denote an arbitrary random variable, is it true that $$\frac{\mathbb{E}[f(X,\mu)]}{\mathbb{E}[g(X,\mu)]}$$ also increases with $\mu$?
For my purpose we can assume $f$ and $g$ are differentiable, and that we can change order of differentiation and expectation in any step of calculation.
Being $E$ a linear and monotone operator, you could prove $E\biggl[\frac{f(X,\mu)}{g(X,\mu)}\biggr]$ is increasing in $\mu$. But as $E$ doesn't function nicely with quotients, you wouldn't be able to prove that $\frac{E[f(X,\mu)]}{E[g(X,\mu)]}$ increases with $\mu$.