Proposition 2

Let $f$ be a function belonging to a Hardy field $\mathscr{K}$.

1 If $f$ is of infinite order relative to $x$, then, for every integer $n > 0$,

(2)

2° If $f$ is of finite order $\mu$ relative to $x$ then, for every $n > 0$,

(3) except if $\mu$ is an integer $\geqslant 0$ and $n > \mu$.

1 If $f$ is of infinite order relative to $x$ one has $\log |f| \gg \log x$, so, since $\log |f|$ and $\log x$ are comparable to any order, $f'/f \gg 1/x$. Put $g = f'/f$; since $g$ is equal to a function in $\mathscr{K}$ on a neighbourhood of $+\infty$ one deduces from $1/g \ll x$, that $g'/g^2 \ll 1$, and so $g'/g \ll g = f'/f$, or again $fg' \ll gf'$. From the relation $f' = fg$ one deduces by differentiating that

or again $f''/f' \sim f'/f$. The same argument, applied to $f^{(n)}$ instead of $f$, shows, by induction on $n$, that $f^{(n)}/f^{(n-1)} \sim f'/f$; whence relation (2).

2 If $f$ is of finite order $\mu$ relative to $x$, and if $\mu \neq 0$, one has $\log |f| \sim \mu \log x$, whence, on differentiating, $f'(x) \sim \mu \frac{f(x)}{x}$; one deduces that $f'$ is of order $\mu - 1$ relative to $x$, which allows one to apply the same argument by induction on $n$ so long as $\mu \neq n$, whence formula (3) when $\mu$ is not an integer $\geqslant 0$ and $< n$.

When $f$ is of integral order $p \geqslant 0$ relative to $x$ one can write $f(x) = x^p f_1(x)$, where $f_1$ is of order 0 relative to $x$. By prop. 2 one has

To evaluate the derivatives of order $n > p$ one may restrict oneself to the case $p = 0$. Then one has $\log |f| \ll \log x$, whence $f'(x)/f(x) \ll 1/x$, in other words $x f'(x) \ll f(x)$; if $f$ is not equivalent to a constant $k \neq 0$ one has, on differentiating this relation (V, p. 232, prop. 7), $x f''(x) + f'(x) \ll f'(x)$, which implies that $x f''(x) \sim -f'(x)$. Taking account of this formula, one sees by induction on $n$ that $f^{(n)}$ is of order $\leqslant -n$ relative to $x$, and that (4)

If $f$ is equivalent to a constant $k \neq 0$ one has $f(x) = k + f_2(x)$ with $f_2 \ll 1$, and it reduces to studying the derivatives of $f_2$.