On a subclass of the class of rapidly varying sequences

We define and study the class KR∞,sKR∞,s which is a proper subclass of the class R∞,sR∞,s of rapidly varying sequences of index of variability ∞. Then, we prove a theorem of the Galambos–Bojanić–Seneta type for this class, as well as a representation theorem in the Karamata sense. We also give some important asymptotic properties and characterizations of sequences in KR∞,sKR∞,s.