Strong Asymptotic Equivalence and Inversion of Functions in the Class Kc

Abstract

In this paper we consider the class of functions K c introduced by W. Matuszewska (1964, Studia Math. 24, 271-279) and W. Matuszewska and W. Orlicz (1965, Studia Math. 26, 11-24). As a main result we describe, in terms of the class K c , when two strictly increasing functions, as well as their inverse functions, are asymptoticaly equivalent in the strong sense. This result also gives a proper characterization of the class K c .

The strong asymptotic equivalence and the generalized inverse

“…It was proved in [13] that, in the class of all continuous and strictly increasing functions from the class A , the claims (a) and (b) of Theorem A hold if and only if f ∈ K * c . Moreover, in [13] there are also noted the next two theorems with only the short ideas of proofs.…”

“…The class ARV is also known to be an important object in [7] and [13]. Its intersection with the class K c gives the class K * c (see [7]).…”

On some extensions of Karamata’s theory and their applications

“…PRV functions and their various applications have been studied by Korenblyum [36], Matuszewska [38], Matuszewska and Orlicz [39], Stadtmüller and Trautner [48], [49], Berman [4,5], Yakymiv [53,54], Cline [19], Klesov et al [33], Djurčić and Torgašev [21], Buldygin et al [9], [11]- [13], [15]- [18]. Note that PRV functions are called regularly oscillating in Berman [4], weakly oscillating in Yakymiv [53] and intermediate regularly varying in Cline [19].…”

“…In a more general setting, problem (D) has been considered in Djurčić and Torgašev [21] and Buldygin et al [9,11]. In these papers, among other questions, PMPV functions are defined (see, [21,Definition 3] and [11, relation (6.2)]).…”

“…In these papers, among other questions, PMPV functions are defined (see, [21,Definition 3] and [11, relation (6.2)]). One of the main properties of PMPV functions is that their quasi-inverse functions are PRV functions and preserve the equivalence of functions.…”