On the improvement of the rate of convergence of the generalized Bieberbach polynomials in domains with zero angles

• Дата: 2012
• Автори: Abdullayev, F. G., Özkartepe, N. P., Абдуллаєв, Ф. Г., Озкартепе, Н. П.
• Формат: Стаття
• Мова: Англійська
• Опубліковано: Institute of Mathematics, NAS of Ukraine 2012
• Онлайн доступ: https://umj.imath.kiev.ua/index.php/umj/article/view/2599
• Назва журналу: Ukrains’kyi Matematychnyi Zhurnal

Let $\mathbb{C}$ be the complex plane, let $\overline{\mathbb{C}} = \mathbb{C} \bigcup \{\infty\}$, let $G \subset \mathbb{C}$ be a finite Jordan domain with $0 \in G$, let $L := \partial G$, let $\Omega := \overline{\mathbb{C}} \ \overline{G}$, and let $w = \varphi(z)$ be the conformal mapping of $G$ onto a disk $B(0, \rho) := \{w : \; |w | < \rho_0\}$ normalized by $\varphi(0) = = 0,\; \varphi'(0) = 1$, where $\rho_0 = \rho_0 (0, G)$ is the conformal radius of $G$ with respect to 0. Let $\varphi \rho(z) := \int^z_0 [\varphi'(\zeta)]^{2/p}d\zeta$ and let $\pi_{n,p}(z)$ be the generalized Bieberbach polynomial of degree $n$ for the pair $(G, 0)$ that minimizes the integral $\int\int_G|\varphi'(z) - P'_n(z)|^p d \sigma_z$ in the class of all polynomials of degree $\text{deg} P_n \leq n$ such that $P_n(0) = 0$ and $P'_n(0) = 1$. We study the uniform convergence of the generalized Bieberbach polynomials $\pi_{n,p}(z)$ to $\varphi \rho(z)$ on $\overline{G}$ 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.