On Convergence Rates of Some Limits

Abstract

In 2019 Seneta has provided a characterization of slowly varying functions L in the Zygmund sense by using the condition, for each y > 0 , x L ( x + y ) L ( x ) − 1 → 0 as x → ∞ . Very recently, we have extended this result by considering a wider class of functions U related to the following more general condition. For each y > 0 , r ( x ) U ( x + y g ( x ) ) U ( x ) − 1 → 0 as x → ∞ , for some functions r and g. In this paper, we examine this last result by considering a much more general convergence condition. A wider class related to this new condition is presented. Further, a representation theorem for this wider class is provided.

Analizing a Seneta's conjecture by using the Williamson transform

“…Nowadays, Kevei in [Kevei(2021)] presented an extension of such a conjecture and its proof. Coincidentally, we have been working on such a subject as a part of other researches related to results shown by [Seneta(2019)], see [Omey and Cadena(2020a)] and [Omey and Cadena(2020b)]. Unlike Kevei's proofs, ours are based on mainly the Williamson transform.…”