New results on the order of functions at infinity

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 $\mathcal{M}(\pm\infty)$.