The strong asymptotic equivalence and the generalized inverse

We discuss the relationship between the strong asymptotic equivalence relation and the generalized inverse in the class $\mathscr A$ of all nondecreasing and unbounded functions, defined and positive on a half-axis $[a,+\infty)$ ($a>0$). In the main theorem, we prove a proper characterization of the function class $IRV\cap\mathscr A$, where $IRV$ is the class of all $\mathscr O$-regularly varying functions (in the sense of Karamata) having continuous index function.