ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

8. Analysis

1991 MS Classification: 30C55 Kayumov, Il’giz, Kazan University, Russia Asymptotical behaviour of logarithmic coefficients of univalent functions Let S be the class of all univalent in E = {|z| < 1} functions such that f(0) = f′(0) − 1 = 0. Let log f′(z) = ∑_{n=1}^{∞} a_{n}(f)z^{n}. Some estimates of coefficients a_{n}(f) were considered in papers[1], [2]. We prove that

max_{f∈S} \overline{lim_{n→∞}}|a_{n}(f)| = \overline{lim_{n→∞}} max_{f∈S} |a_{n}(f)| = A.

In addition, we show that M = \{f ∈ S : \overline{lim}_{n→∞}|a_{n}(f)| = A\} is dense in S with respect to the pointwise convergence. Reference: 1. Leung Y. K., Shober G., Some coefficient problems and applications, Pacif. J. Math. 145 (1990), N 1, 71–95

2. Avhadiev F. G., Kayumov I. R., Estimates for Bloch functions and their generalization, Complex Variables 29 (1996), 193–201.