一类近于凸调和映照的凸像半径与Pre-Schwarz导数的估计

数学物理学报 ›› 2016, Vol. 36 ›› Issue (6): 1057-1066.

一类近于凸调和映照的凸像半径与Pre-Schwarz导数的估计

漆毅, 石擎天

北京航空航天大学数学与系统科学学院 & LMIB 北京 100191

收稿日期:2016-02-27 修回日期:2016-08-13 出版日期:2016-12-26 发布日期:2016-12-26
通讯作者: 石擎天 E-mail:shiqingtian2013@gmail.com
基金资助:
国家自然科学基金（11371045）、中央高校基础科研基金（YWF-14-SXXY-008）和安徽省高等学校自然科学研究项目（KJ2015A323）资助

Estimations of Convexity Radius and the Norm of Pre-Schwarz Derivative for Close to Convex Harmonic Mappings

Qi Yi, Shi Qingtian

School of Mathematics and Systems Science & LMIB, Beihang University, Beijing 100191

Received:2016-02-27 Revised:2016-08-13 Online:2016-12-26 Published:2016-12-26
Supported by:
Supported by the NSFC (11371045),the Fundamental Research Funds for the Central University (YWF-14-SXXY-008) and the Natural Science Foundation of the Education Department of Anhui Province (KJ2015A323)

摘要/Abstract

摘要：

对任意给定的α∈[0，1），对单位圆盘D上规范化的保向调和映照类H的一个近于凸子类
P⁰（α）={f=h+g∈H：R{h'（z）-α}>|g'（z）|，z∈D，g'（0）=0}
的性质进行了研究，如P⁰（α）类的凸像和星象半径估计、偏差定理、像域面积的估计、拟共形性，其中得到的凸像和星象半径估计值改进了文献[8-9]中相应结果.此外，对包含P⁰（α）的稳定单叶调和映照类（SHU）的Pre-Schwarz导数进行了考虑，得到了精确的上界估计.

关键词: 近于凸函数, 凸像和星象半径, 偏差定理, 拟共形映照, Pre-Schwarz导数

Abstract:

In this paper, the following subclass of H
P⁰(α)={f=h+g∈H:Re{h'(z)-α}>|g'(z)|, z∈D and g'(0)=0}
is studied, where α∈[0,1) and H is the class of all normalized sense preserving harmonic mappings defined in the unit disk D. Estimations of the convexity and starlike radii of P⁰(α) are given, which improve the relative results in[8, 9]. A distortion theorem and a lower bound of |f(D)| for all f∈P⁰(α) are obtained. The upper bound of Pre-Schwarzian norms of functions in a subclass of SHU containing P⁰(α) is estimated and the quasiconformality is discussed also.

Key words: Close to convex functions, Convexity and starlike radii, Distortion theorem, Quasiconformal mapping, Pre-Schwarz derivative

中图分类号:

O174.55

漆毅, 石擎天. 一类近于凸调和映照的凸像半径与Pre-Schwarz导数的估计[J]. 数学物理学报, 2016, 36(6): 1057-1066.

Qi Yi, Shi Qingtian. Estimations of Convexity Radius and the Norm of Pre-Schwarz Derivative for Close to Convex Harmonic Mappings[J]. Acta mathematica scientia,Series A, 2016, 36(6): 1057-1066.

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