Exercise 3, Appendix 0

Let $f$ and $g$ be two (H) functions such that $f/g$ is of order 0 relative to $l_m(x)$; show that if $g$ is not of order 0 relative to $l_m(x)$ one has $f'/g' \sim f/g$ (compare $\log |f|$ and $\log |g|$).