Let $f$ be a increasing and bijective function defined on $(0,\infty)$. Suppose also that $\lim_{r\to 0}f(r)=0$ and $\lim_{r\to \infty}f(r)=\infty$ and $f(uv)\geq C f(u)f(v)$ for some $C>0$. I want to show that the function $$ h(r)=\frac{f(r^{-a})}{f(r^{-b})}, $$ where $0<b<a$ is almost-decreasing, i.e., $h(r_1)\geq Ch(r_2)$, for some constant $C>0$, when $r_1<r_2$.
I only find that $$ h(r)=\frac{f(r^{-a})}{f(r^{-b})}=\frac{f(r^{-a})}{f(r^{a-b}r^{-a})}\leq \frac{f(r^{-a})}{f(r^{a-b})f(r^{-a})}=\frac{1}{f(r^{a-b})}, $$ which implies that $\lim_{r\to \infty}h(r)=0$. With using this fact can we say anything about monotonocity of $h$?