Suppose $f,g,h$ are real functions defined on a neighborhood of $\infty$ such that $f\circ g=\Theta(f\circ h)$. Under what conditions on $f$ does it follow that $g=\Theta(h)$?
For instance, it appears that when $f$ is bilipschitz, the result follows. Does it follow for any weaker condition on $f$?