ON THE UNIFORM CONVERGENCE OF THEGENERALIZED BIEBERBACH POLYNOMIALS INREGIONS WITH K-QUASICONFORMAL BOUNDARY

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 $p > 2 - \frac{{K^2 + 1}}{{2K^4 }}$ .

Bieberbach猜想的一个应用及S族的一个性质

运用Bieberbach猜想及级数收敛的定义获得了S族函数及其导函数的模有上界的简洁证明。同时运用面积原理与极限的定义证明了S族函数在原点附近具有保二维测度微小变化的性质。

A proof of a weak version of the Bieberbach conjecture in several complex variables

In this paper, the sharp estimates of all homogeneous expansions for a subclass of starlike mappings on the unit ball in complex Banach spaces are first established. Meanwhile, the sharp estimates of all homogeneous expansions for the above generalized mappings on the unit polydisk in Cnare also obtained. Our results show that a weak version of the Bieberbach conjecture in several complex variables is proved, and the obtained conclusions reduce to the classical results in one complex variable.

Uniform convergence of the p-Bieberbach polynomials in domains with zero angles

Let G C C be a simply connected domain whose boundary L ：= G is a Jordan curve and 0 ∈ G. Let w = φ（z） be the conformal mapping of G onto the disk B（0, r0） ：= {w ： ｜w｜ 〈 r0）, satisfying φ0（0） = 0, φ＇t（0） = 1. We consider the following extremal problem for p 〉 0：∫∫G｜φ＇（z）-P＇n（z）｜Pdσz→min in the class of all polynomials Pn（z） of degree not exceeding n with Pn（0） = 0, P＇n （0）=- 1. The solution to this extremal problem is called the p-Bieberbach polynomial of degree n for the pair （G, 0）. We study the uniform convergence of the p-Bieberbach polynomials Bn,p（z） to the φ（z） on G^- with interior and exterior zero angles determined depending on the properties of boundary arcs and the degree of their ＂touch＂.

关于De Branges定理的注记

假设f(z)=z+a^2z^2+…在单位圆盘D内解析和单叶,De Barnges证明了单叶函数的Bieberach猜测。在这篇文章中,我们改进了De Branges定理。

L.de Branges定理的另一证明

L.de Brngcs于1984年证明了Bicberach猜想,其中关键部分的证明相当复杂,并且应用了特殊函数中的一些知识。本文用简单方法证明L.de Brangcs定理k【7的情形。

C^n中单位多圆柱上一类星形映照第五项齐次展开式的精确估计

本文主要在Cn中单位多圆柱上给出一类正规化双全纯星形映照第五项齐次展开式的精确估计,其结果表明在特定的条件下关于第五项齐次展开式的多复变数Bieberbach猜想成立,它是多复变数已有结果的深化,并能回到单复变数的相应结论.

C^n中单位多圆柱上一类α次星形映照第四项齐次展开式的精确估计

主要在Cn中单位多圆柱上建立一类α次星形映照第四项齐次展开式的精确估计,所得结论是已有结果的进一步深化,且显示在一定的限制条件下,多复变数中关于α次星形映照第四项齐次展开式的Bieberbach猜想成立.

C^n中单位多圆柱上一类α型螺形映射齐次展开式第四项的精确估计

本文主要给出C^n中单位多圆柱上一类正规化双全纯螺形映射第四项齐次展开式的精确估计,所得结果揭示在一定的限制条件下关于α型螺形映射齐次展开式第四项的多复变数Bieberbach猜想成立,从而推广了多复变数已有的结果.

多复变数一类α次星形映射齐次展开式各项的精细估计

本文首先给出复Banach空间单位球上一类α次星形映射齐次展开式各项的精细估计,特别当这些映射又是k折对称映射时,估计还是精确的.其次建立C^n中单位多圆柱上上述推广映射齐次展开式各项的精细估计,同样当这些映射又是k折对称映射时,估计仍是精确的.由此证明了多复变数中关于α次星形映射的弱Bieberbach猜想,且所得到的估计都能回到单复变数的情形.

The Discrete Elementary Groups

This paper discusses discrete elementary subgroups in the Mobius groupM■.With a help of geometry,it is proved that these groups are isomorphic to either agroup extension of Z by a finite subgroup of SO(n) or an extension of a finite group by afree Abelian group of rank k≤n.

Fekete and Szeg problem in one and higher dimensions

In this paper, we establish the Fekete and Szego inequality for a class of holomorphic functions in the unit disk, and then we extend this result to a class of holomorphic mappings on the unit ball in a complex Banach space or on the unit polydisk in Cn.