On factorization representations for Avakumović--Karamata functions with nondegenerate groups of regular points

Abstract

Avakumovic-Karamata functions f are generalized regularly varying functions (so--called ORV functions) such that f*(λ)= limsup x →∞f(λx)/f(x) is finite for all λ>0. In this paper, we investigate classes of ORV functions with "nondegenerate groups of regular points", that is, having points λ≥1, for which f*(λ) exists as a positive and finite limit (instead of limsup) on a nontrivial subgroup of the positive real axis. Certain factorization representations, characterizations and uniform convergence theorems are proved, describing both the structure of ORV functions f as well as that of their limit functions f*. Some well-known results from regular variation theory are covered by this general approach.