On the improvement of the rate of convergence of the generalized Bieberbach polynomials in domains with zero angles

• Main Authors: Abdullayev, F.G., Özkartepe, N.P.
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2012
• Series: Український математичний журнал
• Subjects: Статті
• Online Access: http://dspace.nbuv.gov.ua/handle/123456789/164421

Let ℂ be the complex plane, let C¯=C∪{∞}, let G ⊂ ℂ be a finite Jordan domain with 0 ∈ G; let L := ∂G; let Ω := C¯∖G¯, and let w = φ(z) be a conformal mapping of G onto a disk B(0, ρ₀) := {w:|w|<ρ₀} normalized by the conditions φ(z) = 0 and φ′(0)=1, where ρ₀ = ρ₀(0, G) is the conformal radius of G with respect to 0. Let φp(z):=∫₀z[φ′(ζ)]2/pdζ and let π n,p (z) be the generalized Bieberbach polynomial of degree n for the pair (G, 0) that minimizes the integral ∬G∣∣φ′p(z)−P′n(z)∣∣pdσz in the class of all polynomials of degree deg Pn ≤ n such that Pn(0) = 0 and P′n(0)=1. We study the uniform convergence of the generalized Bieberbach polynomials π n,p (z) to φ p (z) on G¯ with interior and exterior zero angles determined depending on properties of boundary arcs and the degree of their tangency. In particular, for Bieberbach polynomials, we obtain improved estimates for the rate of convergence in these domains.