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In this chapter, we will study some characteristics of the functional transformation K, which maps the class of O-regularly varying functions (see Bingham et al. Regular Variation, 1987) into the class of positive functions on the interval \((0,+\infty )\) defined by \(\displaystyle K(f)=k_f, \) where \(\displaystyle k_f(\lambda )=\limsup _{x\to +\infty } \frac {f(\lambda x)}{f(x)}\) , \(\lambda \in (0,+ \infty )\) , and the function f belongs to the class of O-regularly varying functions in the sense of Karamata (see Aljančić and Arandjelović, Publ Inst Math (Beograd) 22(36):5–22, 1977). We will also study some properties of that operator K if its domain is the class of O-regularly varying sequences in the sense of Karamata, given by \(\displaystyle K(c)=k_c, \) where \(\displaystyle k_c(\lambda )=\limsup _{n\to +\infty } \frac {c_{[\lambda n]}}{c_n}\) , \(\lambda \in (0, +\infty )\) , and sequence \(c=(c_n)\in ORV_c\) (Djurčić et al., On theorems of Galambos-Bojanić-Seneta type, 2022).

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Classical Karamata Theory of Regular Variability and the Index Function Operator

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  • 发表日期：

    2025年

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  • Danica Fatić,
  • Dragan Djurčić

摘要

In this chapter, we will study some characteristics of the functional transformation K, which maps the class of O-regularly varying functions (see Bingham et al. Regular Variation, 1987) into the class of positive functions on the interval \((0,+\infty )\) defined by \(\displaystyle K(f)=k_f, \) where \(\displaystyle k_f(\lambda )=\limsup _{x\to +\infty } \frac {f(\lambda x)}{f(x)}\) , \(\lambda \in (0,+ \infty )\) , and the function f belongs to the class of O-regularly varying functions in the sense of Karamata (see Aljančić and Arandjelović, Publ Inst Math (Beograd) 22(36):5–22, 1977). We will also study some properties of that operator K if its domain is the class of O-regularly varying sequences in the sense of Karamata, given by \(\displaystyle K(c)=k_c, \) where \(\displaystyle k_c(\lambda )=\limsup _{n\to +\infty } \frac {c_{[\lambda n]}}{c_n}\) , \(\lambda \in (0, +\infty )\) , and sequence \(c=(c_n)\in ORV_c\) (Djurčić et al., On theorems of Galambos-Bojanić-Seneta type, 2022).