Let G be a finite domain in the complex plane with K-quasicon formal boundary, z 0 be an arbitrary fixed point in G and p>0. Let π(z) be the conformal mapping from G onto the disk with radius r 0>0 and centered...Let G be a finite domain in the complex plane with K-quasicon formal boundary, z 0 be an arbitrary fixed point in G and p>0. Let π(z) be the conformal mapping from G onto the disk with radius r 0>0 and centered at the origin 0, normalized by ?(z 0) = 0 and ?(z 0) = 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,z 0) 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 (z 0) = 0 and P′ n (z 0) = 1. In this work we prove the uniform convergence of the generalized Bieberbach polynomials π n,p (z) to ? p (z) on $\bar G$ in case of $p > 2 - \frac{{K^2 + 1}}{{2K^4 }}$ .展开更多
文摘Let G be a finite domain in the complex plane with K-quasicon formal boundary, z 0 be an arbitrary fixed point in G and p>0. Let π(z) be the conformal mapping from G onto the disk with radius r 0>0 and centered at the origin 0, normalized by ?(z 0) = 0 and ?(z 0) = 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,z 0) 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 (z 0) = 0 and P′ n (z 0) = 1. In this work we prove the uniform convergence of the generalized Bieberbach polynomials π n,p (z) to ? p (z) on $\bar G$ in case of $p > 2 - \frac{{K^2 + 1}}{{2K^4 }}$ .
