On the Uniform Convergence of the Generalized Bieberbach Polynomials in Regions with K-Quasiconformal Boundary.

Let G be a finite domain in the complex plane with K-quasicon formal boundary, z₀ be an arbitrary fixed point in G and p>0. Let π( z) be the conformal mapping from G onto the disk with radius r₀>0 and centered at the origin 0, normalized by ϕ( z₀) = 0 and ϕ( z₀) = 1. Let us set $$\varphi _p \left( z \right): = \int_{x_0 }^x {\left[ {\phi \left( \zeta \right)} \right]^{2/8} } d\zeta $$ , and let π _n,p( z) be the generalized Bieberbach polynomial of degree n for the pair ( G,z₀) that minimizes the integral $$\iint\limits_c {\left| {\varphi _p \left( z \right) - P_x^1 (z)} \right|^p d0_x }$$ in the class $$\mathop \prod \limits_n $$ of all polynomials of degree ≤ n and satisfying the conditions P _n( z₀) = 0 and P′ _n( z₀) = 1. In this work we prove the uniform convergence of the generalized Bieberbach polynomials π _n,p( z) to ϕ _p( z) on $$\bar G$$ in case of $$p > 2 - \frac{{K^2 + 1}}{{2K^4 }}$$ . [ABSTRACT FROM AUTHOR]