TITLE:
Remarks on Different Approaches to the Theory of Higher-Order Types of Asymptotic Variation
AUTHORS:
Antonio Granata
KEYWORDS:
Higher-Order Types of Asymptotic Variation, Hardy’s Approach, Karamata’s Approach, Bourbaki-Dieudonné’s Approach, Zygmund’s Approach
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.16 No.1,
January
27,
2026
ABSTRACT: As a completion of the previously published theory of higher-order types of asymptotic variation, we point out some remarks about different approaches to construct such an advanced and demanding theory, because there are various (non-equivalent) approaches to the basic concepts of types of asymptotic variation, i.e., the concepts not involving derivatives. After listing the five known approaches to the concept of regular variation and their mutual relationships, we show that, as far as the higher-order theory is concerned, there is essentially “only one way” to construct such a theory for differentiable functions.