The weak and strong asymptotic equivalence relations and the generalized inverse*

Abstract

We discuss the relationship between the weak and strong asymptotic equivalence relations and the generalized inverse in the class A of all nondecreasing unbounded positive functions on a half-axis [a,+∞) (a > 0). As a main result, we prove a proper characterization of the functional class R ∞ ∩ A, where R ∞ is the class of all rapidly varying functions. Also, we prove a characterization of the functional class PI * ∩ A.