Classical Karamata Theory of Regular Variability and the Index Function Operator

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 , + ∞ ) $$(0,+\infty )$$ defined by K ( f ) = k f , $$\displaystyle K(f)=k_f, $$ where k f ( λ ) = lim sup x → + ∞ f ( λx ) f ( x ) $$\displaystyle k_f(\lambda )=\limsup _{x\to +\infty } \frac {f(\lambda x)}{f(x)}$$ , λ ∈ ( 0 , + ∞ ) $$\lambda \in (0,+ \infty )$$ , 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 K ( c ) = k c , $$\displaystyle K(c)=k_c, $$ where k c ( λ ) = lim sup n → + ∞ c [ λn ] c n $$\displaystyle k_c(\lambda )=\limsup _{n\to +\infty } \frac {c_{[\lambda n]}}{c_n}$$ , λ ∈ ( 0 , + ∞ ) $$\lambda \in (0, +\infty )$$ , and sequence c = ( c n ) ∈ OR V c $$c=(c_n)\in ORV_c$$ (Djurčić et al., On theorems of Galambos-Bojanić-Seneta type, 2022).