| Title:
|
Uniform convergence of the generalized Bieberbach polynomials in regions with zero angles (English) |
| Author:
|
Abdullayev, F. G. |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
51 |
| Issue:
|
3 |
| Year:
|
2001 |
| Pages:
|
643-660 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $C$ be the extended complex plane; $G\subset C$ a finite Jordan with $ 0\in G$; $w=\varphi (z)$ the conformal mapping of $G$ onto the disk $ B\left( {0;\rho _{0}}\right):={\left\rbrace {w\:{\left| {w}\right| }<\rho _{0}} \right\lbrace }$ normalized by $\varphi (0)=0$ and ${\varphi }^{\prime }(0)=1$. Let us set $\varphi _{p}(z):=\int _{0}^{z}{{\left[ {{\varphi } ^{\prime }(\zeta )}\right] }^{{2}/{p}}}\mathrm{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}^{\prime }(z)-{P}_{n}^{\prime }(z)}\right| }}^{p}\mathrm{d}\sigma _{z}$ in the class of all polynomials of degree not exceeding $\le n$ with $P_{n}(0)=0$, ${P}_{n}^{\prime }(0)=1$. In this paper we study the uniform convergence of the generalized Bieberbach polynomials $\pi _{n,p}(z)$ to $\varphi _{p}(z)$ 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. (English) |
| Keyword:
|
conformal mapping |
| Keyword:
|
Quasiconformal curve |
| Keyword:
|
Bieberbach polynomials |
| Keyword:
|
complex approximation |
| MSC:
|
30C10 |
| MSC:
|
30C30 |
| MSC:
|
30C70 |
| MSC:
|
30E10 |
| idZBL:
|
Zbl 1079.30506 |
| idMR:
|
MR1851553 |
| . |
| Date available:
|
2009-09-24T10:45:57Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127675 |
| . |
| Reference:
|
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| Reference:
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