Some properties of asymptotic quasi-inverse functions and their applications I

Abstract

Abstract. We introduce the notions of asymptotic quasi-inverse functions and asymptotic inverse functions as weaker versions of (quasi-)inverse functions, and study their main properties. Asymptotic quasi-inverse functions exist in the class of socalled pseudo-regularly varying (PRV) functions, i.e. functions preserving the asymptotic equivalence of functions and sequences. On the other hand, asymptotic inverse functions exist in the class of so-called POV functions, i.e., functions with positive order of variation. In this paper, we obtain some new results about PRV and POV functions. Some further properties of asymptotic (quasi-)inverse functions as well as some applications will be discussed in Part II of this paper to appear in no. 71 of this journal.