On the improvement of the rate of convergence of the generalized Bieberbach polynomials in domains with zero angles
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 the conformal mapping of G onto a disk B(0, ρ) := {w : |w | < ρ_0} normalized by φ(0) = = 0, φ’(0) = 1, where ρ_0 = ρ_0 (0, G) is the conformal radius of G with respect to 0. Let φ ρ(z) := ^z_0 [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’(z) - P’_n(z)|^p d σ_z in the class of all polynomials of degree deg P_n ≤ n such that P_n(0) = 0 and P’_n(0) = 1. We study the uniform convergence of the generalized Bieberbach polynomials π_n,p(z) to φ ρ(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 better estimates for the rate of convergence in these domains., number=5, journal=Ukrains’kyi Matematychnyi Zhurnal, author=Abdullayev, F. G. and Özkartepe, N. P., year=2012, month=May, pages=582–596