Uniform Convergence of the Generalized Bieberbach Polynomials in Regions with Zero Angles

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 $B(0;\varrho _0 ): = \{ w:|w| < \varrho _0 \}$ normalized by $\varphi (0) = 0 {\text{and}} \varphi '(0) = 1$. Let us set $\varphi _p (z): = \int_0^z {\left[ {\varphi '(\zeta )} \right]} ^{{\raise0.7ex\hbox{$2$} \!\mathord{\left/ {\vphantom {2 p}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$p$}}} {\text{d}}\zeta$, and let $\pi _{n,p} (z)$ be the generalized Bieberbach polynomial of degree n for the pair (G,0), which minimizes the integral $\iint\limits_G {\left| {\varphi '_p (z) - P'_n (z)} \right|^p {\text{d}}\sigma _{\text{z}} }$ in the class of all polynomials of degree not exceeding ≤ n with $P_n (0) = 0,{\text{ }}P'_n (0) = 1$. In this paper we study the uniform convergence of the generalized Bieberbach polynomials $\pi _{n,p} (z){\text{ to }}\varphi _p (z){\text{ on }}\overline G$ with interior and exterior zero angles and determine its dependence on the properties of boundary arcs and the degree of their tangency.