Bulletin of the Belgian Mathematical Society - Simon Stevin

Convergence of Bieberbach polynomials inside domains of the complex plane

F.G. Abdullayev, M. Küçükaslan, and T. Tunç

Full-text: Open access

Abstract

Let $G\subset C $ be a finite Jordan domain, $z_{0}\in G;$ $B\Subset G$ be an arbitrary closed disk with $z_{0}\in B,$ and $w=\varphi (z,z_{0})$ be the conformal mapping of $G$ onto a disk $\{w:\left| w\right| <r\}$ normalized by $\varphi (z_{0},z_{0})=0$, $\varphi ^{\prime }(z_{0},z_{0})=1$ . It is well known that the Bieberbach polynomials $\{\pi _{n}(z,z_{0})\}$ for the pair $(G,z_{0})$ converge uniformly to $\varphi (z,z_{0})$ on compact subsets of the Jordan domain $G.$ In this paper we study the speed of $\left\| \varphi -\pi _{n}\right\| _{C(B)}\rightarrow 0,$ $n\rightarrow \infty ,$ in domains of the complex plane with a complicated boundary structure.

Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 13, Number 4 (2006), 657-671.

Dates
First available in Project Euclid: 16 January 2007

Permanent link to this document
https://projecteuclid.org/euclid.bbms/1168957342

Digital Object Identifier
doi:10.36045/bbms/1168957342

Mathematical Reviews number (MathSciNet)
MR2300622

Zentralblatt MATH identifier
1130.30007

Subjects
Primary: 30C30: Numerical methods in conformal mapping theory [See also 65E05] 30E10: Approximation in the complex domain 30C70: Extremal problems for conformal and quasiconformal mappings, variational methods

Keywords
Conformal mapping Quasiconformal curve Bieberbach polynomials Complex approximation

Citation

Küçükaslan, M.; Tunç, T.; Abdullayev, F.G. Convergence of Bieberbach polynomials inside domains of the complex plane. Bull. Belg. Math. Soc. Simon Stevin 13 (2006), no. 4, 657--671. doi:10.36045/bbms/1168957342. https://projecteuclid.org/euclid.bbms/1168957342


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