Uniform convergence of Bieberbach polynomials in domains with interior zero angles. (Russian. English summary) Zbl 0708.30039

Abdullaev, F. G.

Dokl. Akad. Nauk Ukr. SSR, Ser. A 1989, No. 12, 3-5 (1989).

Let G be a finite domain in \({\mathbb{C}}\) bounded by a Jordan curve L. Given a fixed point \(z_ 0\) in G let f be the conformal map of G onto a disk \(\{| w| <r\}\) such that \(f(z_ 0)=0\) and \(f'(z_ 0)=1\). Let \(B_ n\) be the unique polynomial of degree \(\leq n\) (Bieberbach polynomial) for which the integral \[ \iint_{G}| P'(z)|^ 2 d\sigma \] is minimal in the class of all polynomials P of degree \(\leq n\) with \(P(z_ 0)=0\), \(P'(z_ 0)=1.\)
The author gives (without proofs) estimates of the expressions \(\sup_{z\in G}| f(z)-B_ n(z)|\) for some classes of domains G such that \(L=\partial G\) is a union of quasiconformal arcs \(L_ 1,...,L_ m\) meeting under zero inner angles. As a special case of his results the author obtains the following S. N. Mergelyan’s theorem [Trudy Mat. Inst. Steklov. 37, 92 p. (1951; Zbl 0045.353)]: If \(L=\partial G\) is a Jordan curve with continuously turning tangent then for every \(\epsilon >0\) \[ \sup_{z\in G}| f(z)-B_ n(z)| \leq C(\epsilon)n^{\epsilon -}. \]

Reviewer: J.Siciak

MSC:

30E10	Approximation in the complex plane
30C20	Conformal mappings of special domains

Keywords:

conformal; Bieberbach polynomial

Citations:

Zbl 0045.353