Uniform convergence of the generalized Bieberbach polynomials in regions with zero angles

ABDULLAYEV F.

Czechoslovak Mathematical Journal, vol.51, no.3, pp.643-660, 2001 (SCI-Expanded, Scopus) identifier

  • Publication Type: Article / Article
  • Volume: 51 Issue: 3
  • Publication Date: 2001
  • Doi Number: 10.1023/a:1013796308878
  • Journal Name: Czechoslovak Mathematical Journal
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.643-660
  • Keywords: Bieberbach polynomials, Complex approximation, Conformal mapping, Quasiconformal curve
  • Uşak University Affiliated: No

Abstract

Let C be the extended complex plane; G ⊂ C a finite Jordan with 0 ∈ G; w = φ(z) the conformal mapping of G onto the disk B (0; e0) := {w: |w| < e0} normalized by φ(0) = 0 and φ(0) = 1. Let us set φp(z) := ∫0z2 [φ(ζ)] 2/pdζ, and let πn,p(z) be the generalized Bieberbach polynomial of degree n for the pair (G, 0), which minimizes the integral ∫∫G|φp(z) - P′n(z)|p dσz in the class of all polynomials of degree not exceeding ≤ n with Pn(0) = 0, P′n(0) = 1. In this paper we study the uniform convergence of the generalized Bieberbach polynomials πn,p(z) to φp(z) on G with interior and exterior zero angles and determine its dependence on the properties of boundary arcs and the degree of their tangency.