Exercise 14, Appendix 0

Deduce from exerc. 13 that if $f$ is an (H) function such that $\log f(e^x)$ is of order $< 1$ relative to $x$, the function $f(x f(x))$ is equivalent to a function of the form $e^{g(x)}$, where $g$ is a rational function of $x, \log f(x)$, and of a certain number of derivatives of this last function.