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).