Definition 5

Let $g$ be a function in $\mathscr{H}(\mathfrak{F}, \mathbf{R})$, strictly positive on a set in $\mathfrak{F}$, and such that $\lim_{\mathfrak{F}} g = 0$ or $\lim_{\mathfrak{F}} g = +\infty$. One says that a function $f \in \mathscr{H}(\mathfrak{F}, \mathbf{R})$ is of order $\rho$ (finite or infinite) relative to $g$ if $\lim_{\mathfrak{F}} \log(|f|)/\log g = \rho$.

Note that if $f$ is of order $\rho$ relative to $g$ then $f$ is of order $-\rho$ relative to $1/g$; one therefore need treat only the case where $g(x)$ tends to $+\infty$ along $\mathfrak{F}$.