Exercise 4, Appendix 0

Let $\mathscr{K}$ be a Hardy field such that, for every function $f \in \mathscr{K}$ not equivalent to a constant, and of order 0 relative to $l_{m-1}(x)$, there exist a constant $k$ and a rational integer $r$ such that $f(x) \sim k(l_m(x))^r$.

a) Let $z$ be any function in $\mathscr{K}$, not identically zero on a neighbourhood of $+\infty$. Show that every function $g$ in the Hardy field $\mathscr{K}(\log |z|)$ not equivalent to a constant, and of order 0 relative to $l_m(x)$, is equivalent to a function of the form $k(l_{m+1}(x))^r$ ($k$ constant, $r$ a rational integer). (First consider the case where $g$ is a polynomial of degree $p$ in $\log |z|$, with coefficients in $\mathscr{K}$, and argue by induction on $p$, using exerc. 3; if $g$ is a rational function of $\log |z|$, with coefficients in $\mathscr{K}$, argue by induction on the degree of the numerator, using exerc. 3.)

b) Show that every function in $\mathscr{K}(l_{m+1}(x))$ is equivalent to a function of the form $f(x)(l_{m+1}(x))^r$ where $f \in \mathscr{K}$ and $r$ is a rational integer.

c) Let $z$ be any function in $\mathscr{K}$. Show that every function $g$ of the Hardy field $\mathscr{K}(e^z)$ of order 0 relative to $l_m(x)$ is equivalent to a constant. (First consider the case where $g = u^q P(u)$, where $u = e^z$, $q$ is a rational integer, and $P(u)$ is a polynomial in $u$, of degree $p$, with coefficients in $\mathscr{K}$; then argue by induction on $p$, using $a$) and exerc. 3 Pass from there to the general case by using exerc. 3.)

d) Extend the result of a) to the field $\mathscr{K}(u_1, u_2, \ldots, u_s)$, where the $u_k$ are of the form $e^{z_k}$ or $\log |z_k|$, the $z_k$ being functions in $\mathscr{K}$ not identically zero on a neighbourhood of $+\infty$ (argue by induction on $s$).