Uniform convergence of the generalized Bieberbach polynomials in regions with zero angles

F. G. Abdullayev

Czechoslovak Mathematical Journal (2001)

  • Volume: 51, Issue: 3, page 643-660
  • ISSN: 0011-4642

Abstract

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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 ; ρ 0 : = w w < ρ 0 normalized by ϕ ( 0 ) = 0 and ϕ ' ( 0 ) = 1 . Let us set ϕ p ( z ) : = ∫ 0 z ϕ ' ( ζ ) 2 / p d ζ , and let π n , p ( z ) be the generalized Bieberbach polynomial of degree n for the pair ( G , 0 ) , which minimizes the integral ∫ ∫ G ϕ p ' ( z ) - P n ' ( z ) p d σ z in the class of all polynomials of degree not exceeding ≤ n with P n ( 0 ) = 0 , P n ' ( 0 ) = 1 . In this paper we study the uniform convergence of the generalized Bieberbach polynomials π n , p ( z ) to ϕ p ( z ) on G ¯ with interior and exterior zero angles and determine its dependence on the properties of boundary arcs and the degree of their tangency.

How to cite

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Abdullayev, F. G.. "Uniform convergence of the generalized Bieberbach polynomials in regions with zero angles." Czechoslovak Mathematical Journal 51.3 (2001): 643-660. <http://eudml.org/doc/30661>.

@article{Abdullayev2001,
abstract = {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.},
author = {Abdullayev, F. G.},
journal = {Czechoslovak Mathematical Journal},
keywords = {conformal mapping; Quasiconformal curve; Bieberbach polynomials; complex approximation; conformal mapping; quasiconformal curve; Bieberbach polynomial; complex approximation},
language = {eng},
number = {3},
pages = {643-660},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Uniform convergence of the generalized Bieberbach polynomials in regions with zero angles},
url = {http://eudml.org/doc/30661},
volume = {51},
year = {2001},
}

TY - JOUR
AU - Abdullayev, F. G.
TI - Uniform convergence of the generalized Bieberbach polynomials in regions with zero angles
JO - Czechoslovak Mathematical Journal
PY - 2001
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 51
IS - 3
SP - 643
EP - 660
AB - 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.
LA - eng
KW - conformal mapping; Quasiconformal curve; Bieberbach polynomials; complex approximation; conformal mapping; quasiconformal curve; Bieberbach polynomial; complex approximation
UR - http://eudml.org/doc/30661
ER -

References

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