  Recently, new classes of positive and measurable functions,
$\mathcal{M}(\rho)$ and $\mathcal{M}(\pm \infty)$, have been defined in terms
of their asymptotic behaviour at infinity, when normalized by a logarithm
(Cadena et al., 2015, 2016, 2017). Looking for other suitable normalizing
functions than logarithm seems quite natural. It is what is developed in this
paper, studying new classes of functions of the type $\displaystyle
\lim_{x\rightarrow \infty}\log U(x)/H(x)=\rho <\infty$ for a large class of
normalizing functions $H$. It provides subclasses of $\mathcal{M}(0)$ and
$\mathcal{M}(\pm\infty)$.
