Uniform convergence of the p-Bieberbach polynomials in domains with zero angles

Abstract

Let G ⊂ ℂ be a simply connected domain whose boundary L := ∂G is a Jordan curve and 0 ∈ G. Let w = φ(z) be the conformal mapping of G onto the disk B(0, r0) := {w : |w| 0: $$\iint_G {|\phi '(z) - P'_n (z)|^p d\sigma _z \to \min }$$ in the class of all polynomials Pn(z) of degree not exceeding n with Pn(0) = 0, Pn′ (0) = 1. The solution to this extremal problem is called the p-Bieberbach polynomial of degree n for the pair (G, 0). We study the uniform convergence of the p-Bieberbach polynomials Bn,p(z) to the φ(z) on \(\bar G\) with interior and exterior zero angles determined depending on the properties of boundary arcs and the degree of their “touch”.