Proposition 1

Two functions in the same Hardy field are comparable to any order (V, p. 232).

Indeed, if $f$ belongs to a Hardy field $\mathscr{K}$, then for every integer $n > 0$ there exists an interval $[x_0, +\infty[$ on which $f$ is $n$ times differentiable, its $n^{th}$ derivative being equal to a function in $\mathscr{K}$ on this interval. It is therefore enough to show that any two functions $f, g$ of $\mathscr{K}$ are comparable. This is evident if one of the functions is identically zero on a neighbourhood of $+\infty$; one may therefore restrict oneself to the case where they are both strictly positive on a neighbourhood of $+\infty$. But then, for every real number $t$, $f - tg$ is equal to a function in $\mathscr{K}$ on a neighbourhood of $+\infty$, so has constant sign on a neighbourhood of $+\infty$, which proves the proposition (V, p. 217, prop. 9).

One deduces immediately from this proposition that, if a Hardy field $\mathscr{K}$ contains the real constants (as we shall always assume in what follows), and if $f$ and $g$ are any two functions in $\mathscr{K}$ then any two of the functions $e^f, e^g, \log |f|, \log |g|, |f|^{\alpha}, |g|^{\alpha}$ ($\alpha$ an arbitrary real), $\int_a f, \int_a g$ ($a$ being any real number in an interval $[x_0, +\infty[$ where $f$ and $g$ are regulated) are comparable (when they are defined); indeed, any two of these functions belong to a given Hardy field obtained by adjoining them successively to $\mathscr{K}$.

Similarly, every function $f(x)$ in a Hardy field $\mathscr{K}$ is comparable to $x$, since $x$ and $f(x)$ belong to the Hardy field obtained by adjoining $x$ to $\mathscr{K}$. One thus concludes (in particular) that $f$ is comparable to any order to every power $x^{\alpha}$, as well as to $\log x$ and to $e^x$.

One also sees that if $f$ and $g$ belong to the same Hardy field $\mathscr{K}$, if $g(x) > 0$ on an interval $[x_0, +\infty[$, and if $g(x)$ tends to 0 or to $+\infty$ as $x$ tends to $+\infty$, then the order of $f$ relative to $g$ (V, p. 219) is always defined.

Prop. 8 of V, p. 233, is therefore applicable to every function $f$ in a Hardy field, and proves that:

1) if $f$ is of order $+\infty$ relative to $x$ then $\int_a^x f(t)\,dt \sim (f(x))^2/f'(x)$.

2) if $f$ is of order $\mu > -1$ relative to $x$ then $\int_a^x f(t)\,dt \sim \frac{1}{\mu+1} x f(x)$.

3) if $f$ is of order $\mu < -1$ relative to $x$ then $\int_a^{+\infty} f(t)\,dt \sim -\frac{1}{\mu+1} x f(x)$.

4) if $f$ is of order $-\infty$ relative to $x$ then $\int_a^{+\infty} f(t)\,dt \sim -(f(x))^2/f'(x)$.

Further, we have the following proposition: