Lemma 2

Let $a(x)$ and $b(x)$ be two functions belonging to a given Hardy field $\mathfrak{K}$ and $y(x)$ a function satisfying the identity (1) on an interval $[x_0, +\infty[$ where $a$ and $b$ are defined and continuous. If $p(u)$ is a polynomial in $u$ whose coefficients are functions of $x$ belonging to $\mathfrak{K}$, defined and differentiable on $[x_0, +\infty[$, then there exists an interval $[x_1, +\infty[$ on which the function $p(y)$ does not change sign.

The proposition is trivial if $p(u)$ has coefficients which are identically zero on $[x_0, +\infty[$, or if $p(u)$ is of degree 0 in $u$, since any function in $\mathfrak{K}$ is of constant sign on an interval $[x_1, +\infty[$. Suppose that $p(u)$ is of degree $n > 0$; the leading coefficient $c$ of $p(u)$ is then $\neq 0$ on an interval $[\alpha, +\infty[$; one can thus write $p(u) = c(u^n + c_1 u^{n-1} + \cdots + c_n)$ where $c, c_1, c_2, \ldots, c_n$ are functions belonging to $\mathfrak{K}$ and differentiable on $[\alpha, +\infty[$; so it is enough to prove the lemma for $c = 1$. We argue by induction on $n$; one has

where $q(u)$ is a polynomial of degree $\leq n - 1$, with coefficients in $\mathfrak{K}$. By hypothesis the functions $na(x)$ and $q(y(x))$ do not change sign on an interval $[ \beta, +\infty[$; the lemma is thus a consequence of lemma 1.