Consider monotone decreasing functions $f,g:[0,\infty)\longrightarrow[0,1]$, such that $f\le g$, $f(0)=g(0)=1$, both vanishing as $x\longrightarrow\infty$ and integrable. I would like to compare $R_f(p)=\frac{f(p)}{\int_0^\infty f(x)dx}$ with $R_g(p)$. Is it true that there always exist a $p^*\in(0,\infty)$ such that $R_f(p^*)\ge R_g(p^*)$?
My intuition would be that the integral affects the ratio heavilier than the function, at some point, but I am not sure how to go about it formally, and although it is true for most examples I tried, there might be additional hypotheses needed. The function $R_f(p)$ vaguely reminds me of the mean value theorem, but I was not able to conclude anything through it in general. Thank's for any suggestion.