Exercise 1, Appendix 0

Let $\mathscr{K}$ be a Hardy field such that, for every function $f \in \mathscr{K}$ not identically zero on a neighbourhood of $+\infty$ there exists a $\lambda > 0$ such that

(m an integer independent of $f$).

a) Let $u_1, u_2, \ldots, u_p$ be $p$ functions of the form $u_k = \log |z_k|$, where $z_k \in \mathscr{K}$ is not zero on a neighbourhood of $+\infty$. Show that for every function $g$ (not zero on a neighbourhood of $+\infty$) of the Hardy field $\mathscr{K}(u_1, u_2, \ldots, u_p)$ obtained by adjoining the functions $u_k$ ($1 \leq k \leq p$) to $\mathscr{K}$, there exists a number $\mu > 0$ such that

(reduce to the case where $g$ is a polynomial in the $u_k$ with coefficients in $\mathscr{K}$ and argue by induction on $p$, then, for $p = 1$, argue by induction on the degree of the polynomial $g$, proceeding as in lemma 2 of V, p. 248).

b) Let $u_k$ ($1 \leq k \leq p$) be $p$ functions of the form $u_k = \exp(z_k)$ where $z_k \in \mathscr{K}$. Show that for every function $g$ of the Hardy field $\mathscr{K}(u_1, u_2, \ldots, u_p)$ not identically zero on a neighbourhood of $+\infty$ there exists a number $\mu > 0$ such that

(similar method).

c) Deduce that if $f$ is an (H) function having a definition sequence with $n$ terms, and not identically zero on a neighbourhood of $+\infty$, there exists a number $\lambda > 0$ such that