The weak asymptotic equivalence and the generalized inverse

In this paper, we discuss the relationship between the weak asymptotic equivalence relation and the generalized inverse in the class \({\mathcal{A}} \) of all nondecreasing and unbounded functions, defined and positive on a half-axis a,+∞) (a > 0). In the main theorem, we prove a proper characterization of the functional class \({{ORV} \cap \mathcal{A}} \), where ORV is the class of all \({\mathcal{O}} \)-regularly varying functions (in the sense of Karamata).