改进的Roper-Suffridge算子的性质及其偏差

数学物理学报 ›› 2014, Vol. 34 ›› Issue (6): 1611-1618.

改进的Roper-Suffridge算子的性质及其偏差

崔艳艳¹|王朝君¹|刘浩²|朱思峰^1*

1.周口师范学院数学与统计学院河南周口 466001;
2.河南大学数学与信息科学学院开封 475001

收稿日期:2013-10-30 修回日期:2014-08-29 出版日期:2014-12-25 发布日期:2014-12-25
基金资助:
国家自然科学基金(11271359, U1204618)、河南省自然科学基金(2011B110034)和河南省教育厅科学技术研究重点项目(14B110015, 14B110016)资助

The Property and the Distortion |of the Generalized Roper-Suffridge Extension Operators

CUI Yan-Yan¹, WANG Chao-Jun¹, LIU Hao², ZHU Sai-Feng^1*

1.College of Mathematics and Statistics, Zhoukou Normal University, Henan\quad Zhoukou 466001;
2.College of Mathematics and Information Science, Henan University, Kaifeng 475001

Received:2013-10-30 Revised:2014-08-29 Online:2014-12-25 Published:2014-12-25
Supported by:
国家自然科学基金(11271359, U1204618)、河南省自然科学基金(2011B110034)和河南省教育厅科学技术研究重点项目(14B110015, 14B110016)资助

摘要/Abstract

摘要：

进一步推广了Roper-Suffridge算子, 并讨论推广后的算子保持双全纯映照子族的一些性质, 从定义出发证明推广后的算子在Cⁿ
中的单位球Bⁿ上保持α次β 型螺形性及强β 型螺形性, 并作为特殊情况得出推广后的算子在相应域上保持α次星形性及强星形性,
且讨论了推广后的Roper-Suffridge算子的偏差定理.

关键词: Roper-Suffridge算子, α次β 型螺形映照, 强&beta, 型螺形映照, 偏差定理

Abstract:

In this paper, the authors generalize the Roper-Suffridge extension operator, and discuss that the generalized operators preserve some properties of subclasses of biholomorphic mappings. From the definition, they prove the fact that the generalized operators preserve spirallikeness of type β and order α and strongly spirallikeness of type β, and as the special case, they obtain that the generalized operators preserve starllikeness of order α and strongly starllikeness on the corresponding domain, and they also research the distortion of the generalized operators.

Key words: Roper-Suffridge operators, Spirallike mappings of type &beta, and order α, Strongly spirallike mappings of type β, Distortion theorem

中图分类号:

32A30

崔艳艳, 王朝君, 刘浩, 朱思峰. 改进的Roper-Suffridge算子的性质及其偏差[J]. 数学物理学报, 2014, 34(6): 1611-1618.

CUI Yan-Yan, WANG Chao-Jun, LIU Hao, ZHU Sai-Feng. The Property and the Distortion |of the Generalized Roper-Suffridge Extension Operators[J]. Acta mathematica scientia,Series A, 2014, 34(6): 1611-1618.

参考文献

[1] Chuaqui M. Applications of subordination chains to starlike mappings in Cⁿ. Pacif J Math, 1995, 168(1): 33--48

[2] Kohr G, Liczberski P. On strongly starlikeness of order alpha in several complex variables. Glasnik Matem, 1995, 33(53): 185--198

[3] Graham I, Kohr G. Univalent mappings associated with the Roper-Suffridge extension operator. Analyse Math, 2000, 81(1): 331--342

[4] Liu M S, Zhu Y C. On the extension operator in Banach spaces. Advances in Math, 2005, 34(4): 506--508

[5] 刘小松, 刘太顺. Reinhardt域上和复Hilbert空间单位球上推广的Roper-Suffridge算子. 数学年刊, 2005, 26A(5): 721--730

[6] Feng S X, Liu T S. The generalized Roper-Suffridge extension operator. Acta Math Sci, 2008, 28B(1): 63--80

[7] 徐庆华, 刘太顺. 正规化双全纯映照的增长和掩盖定理. 数学年刊, 2009, 30A(2): 213--220

[8] 邹娟. 复Banach空间中的Roper-Suffridge算子. 数学物理学报, 2010, 30A(6): 1582--1591

[9] Liu X S, Feng S X. A remark on the generalized Roper-Suffridge extension operator for spirallike mappings of type
β and order α. Chin Quart J of Math, 2009, 24(2): 310--316

[10] 冯淑霞, 张晓飞, 陈慧勇. 多复变数的抛物星形映射. 数学学报, 2011, 54(3): 467--482

[11] Muir J R. A modification of the Roper-Suffridge extension operator. Comput Methods Funct Theory, 2005, 5(1): 237--251

[12] Muir J R, Suffridge T J. Unbounded convex mappings of the ball in Cⁿ. Trans Amer Math Soc, 2001, 129(11): 3389--3393

[13] Muir J R, Suffridge T J. Extreme points for convex mappings of Bⁿ. J Anal Math, 2006, 98: 169--182

[14] Kohr G. Loewner chains and a modification of the Roper-Suffridge extension operator. Mathematica, 2006, 71(1): 41--48

[15] Muir J R. A class of Loewner chain preserving extension operators. J Math Anal Appl, 2008, 337(2): 862--879

[16] 王建飞, 刘太顺. 全纯映照子族上改进的Roper-Suffridge算子. 数学年刊, 2010, 31A(4): 487--496

[17] 冯淑霞.多复变函数的几类全纯映射族. 合肥: 中国科学技术大学, 2004

[18] Koebe P. "{U}ber die Uniformisierung beliebiger analylischer kurven. Nachr Akad Wiss G\"{o}ttingen, Math Phys Kl, 1907, 1907(1): 191--210

[19] Hamada H, Kohr G. The growth theorem and quasiconformal extension of strongly spirallike mappings of type β.
Complex variables, 2001, 44(4): 281--297