Proposition 10

Let $f$ and $g$ be two functions in $\mathscr{H}(\mathfrak{F}, \mathbf{R})$ which are strictly positive on a set in $\mathfrak{F}$; for every $\alpha$ real and $\neq 0$ the relation $f \asymp g$ (resp. $f \sim g$) is equivalent to $f^\alpha \asymp g^\alpha$ (resp. $f^\alpha \sim g^\alpha$); if $\alpha > 0$ the relation $f \preccurlyeq g$ (resp. $f \ll g$) is equivalent to $f^\alpha \preccurlyeq g^\alpha$ (resp. $f^\alpha \ll g^\alpha$); if $\alpha < 0$ it is equivalent to $f^\alpha \succcurlyeq g^\alpha$ (resp. $f^\alpha \succ g^\alpha$).

The proofs are immediate.

One notes that in $\mathscr{H}(\mathfrak{F}, \mathbf{R})$ the set $I'$ of functions strictly positive on a set in $\mathfrak{F}$ is such that $\Gamma/\mathbf{R}_\infty$ is a multiplicative group $\Gamma_\infty$ in $\mathscr{H}_\infty(\mathfrak{F}, \mathbf{R})$; $\Gamma/\mathbf{R}_0$ is identical to the quotient group of $\Gamma_\infty$ by the subgroup of classes (mod. $\mathbf{R}_\infty$) of logarithmically bounded functions in $\Gamma$; on $\Gamma/\mathbf{R}_0$ the order relation deduced from the relation $f \preccurlyeq g$ by passage to the quotient is compatible with the group structure of $\Gamma/\mathbf{R}_0$ and thus makes it an ordered group.