If $f(x,y)/g(x,y)$ is increasing in $x$ for all $y$, will $\int f(x,y)d\mu(y)/\int g(x,y)d\mu(y)$ be also increasing?

Question

You can assume that the functions and the measure $\mu$ are strictly positive and the integrals are finite.

I am asking because I found this paper that establishes a result that is rather similar. However, I am not sure if the monotonicity I am looking for is obvious, or if it is a special case of their result.

Thank you

Answer 1

In the case that $f,g$ are continuous.

For fixed $k,$ and $x\ge k,$ let $F(x)=\int_k^xf(t)dt$ and $G(x)=\int_k^xg(t)dt.$

Then $F/G$ is increasing on $(k,\infty)$ iff $\;0\le (F/G)'=(Gf-Fg)/G^2\;$ iff $\;0\le Gf-Fg\;$ iff $$\frac {F(x)}{G(x)}\le \frac {f(x)}{g(x)}$$ for $x>k.$

By the Second Mean Value Theorem, there exists $y\in (k,x)$ with $$\frac {F(x)}{G(x)}=\frac {F(x)-F(k)}{G(x)-G(k)}=\frac {F'(y)}{G'(y)}=\frac {f(y)}{g(y)}\le \frac {f(x)}{g(x)}.$$