ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

8. Analysis

1991 MS Classification: 26, 33, 40 Simic, Slavko, Belgrade, Yugoslavia Entire Functions of Finite Order and Regularly Varying Sequences Regularly varying sequences (r. v. s) of index α ∈ R are defined as c_{n} := n^{α}L(n), where L(n), n ∈ N are so-called slowly varying sequences in the sense of Karamata, satisfying: lim_{n} \frac{c_{[λn]}}{c_{n}} = 1 for each λ ∈ R^{+}. Examples

of L(n) : ln^{a}2n, ln^{b}(ln 3n), e^{(ln 3n)^{c}}, e^{\frac{ln 2n}{ln(ln 3n)}} , …; a, b ∈ R, 0 < c < 1. We consider a set of entire functions f(x) := Σ_{k=1}^{∞}a_{k}x^{k} of finite order ρ, 0 < ρ < +∞, with non-negative coefficients, and an operator f̂: f̂(x) := \frac{xf′(x)}{f(x)}, x ∈ R^{+}. In this paper we introduce a method for estimationg the asymptotic behavior of f^{∗}(x) := Σ_{k=1}^{∞}c_{k}a_{k}x^{k}, x → ∞, and, among others, prove: (A: )

If ˆ f̂(x) is bounded from above, then \frac{f^{∗}(x)}{f(x)} ∼ c_{[f̂(x)]}, x → ∞ for any r. v. s

(c_{k}) of negative index. (B: ) If f̂(x) ∈ SR_{ρ} (smoothly varying functions of index ρ), then \frac{f^{∗}(x)}{f(x)} ∼ ρ^{α}c_{[lnf(x)]}, x → ∞ for any r. v. s (c_{k}) of arbitrary index α ∈ R.