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On the Refined Esitmates of All Homogeneous Expansions for a Subclass of Biholomorphic Starlike Mappings in Several Complex Variables

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Chinese Annals of Mathematics,Series B 2021年第6期42卷 909-920页

作者： Xiaosong liu taishun liu School of Mathematics and Statistics Lingnan Normal UniversityZhanjiang 524048GuangdongChina Department of Mathematics Huzhou UniversityHuzhou 313000ZhejiangChina

The refined estimates of all homogeneous expansions for a subclass of biholomorphic starlike mappings are mainly established on the unit ball in complex Banach spaces or the unit polydisk in C^(n) with a unified method.Especially the results are sharp if the above mappings are further k-fold symmetric starlike mappings or k-fold symmetric starlike mappings of orderα.The obtained results unify and generalize the corresponding results in some prior literatures.

关键词： Refined estimates of all homogeneous expansions Starlike mapping Starlike mapping of orderα k-fold symmetric Unified method

The Estimates of All Homogeneous Expansions for a Subclass of ε Quasi-convex Mappings in Several Complex Variables

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Chinese Annals of Mathematics,Series B 2018年第4期39卷 621-632页

作者： Xiaosong liu taishun liu School of Mathematics and Statistics Lingnan Normal University Zhanjiang 524048 Guangdong China. Department of Mathematics Huzhou Teachers College Huzhou 313000 Zhejiang China.

The authors obtain the estimates of all homogeneous expansions for a subclass of ε quasi-convex mappings on the unit ball in complex Banach spaces. Moreover, the estimates of all homogeneous expansions for the above generalized mappings on the unit polydisk in Cnare also obtained. Especially, the above estimates are only sharp for a subclass of starlike mappings, quasi-convex mappings and quasi-convex mappings of type A. The results are the generalization of many known results.

关键词：伪凸估计同类子类 Banach

Sharp Distortion Theorems for a Subclass of Biholomorphic Mappings Which Have a Parametric Representation in Several Complex Variables

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Chinese Annals of Mathematics,Series B 2016年第4期37卷 553-570页

作者： Xiaosong liu taishun liu School of Mathematics and Computation Science Lingnan Normal University Department of Mathematics Huzhou University

In this paper, the sharp distortion theorems of the Fr′echet-derivative type for a subclass of biholomorphic mappings which have a parametric representation on the unit ball of complex Banach spaces are established, and the corresponding results of the above generalized mappings on the unit polydisk in C^n are also given. Meanwhile, the sharp distortion theorems of the Jacobi determinant type for a subclass of biholomorphic mappings which have a parametric representation on the unit ball with an arbitrary norm in C^n are obtained, and the corresponding results of the above generalized mappings on the unit polydisk in C^n are got as well. Thus, some known results in prior literatures are generalized.

关键词：双全纯映射参数表示定理失真多复变量 Frechet导数复巴拿赫空间广义映射

Schwarz Lemma at the Boundary on the Classical Domain of Type Ⅲ

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Chinese Annals of Mathematics,Series B 2020年第3期41卷 335-360页

作者： taishun liu Xiaomin TANG Wenjun ZHANG Department of Mathematics Huzhou UniversityHuzhou 313000ZhejiangChina Department of Mathematics Shenzhen UniversityShenzhen 518060GuangdongChina

Let R_Ⅲ(n) be the classical domain of type Ⅲ with n≥2. This article is devoted to a deep study of the Schwarz lemma on R_Ⅲ(n) via not only exploring the smooth boundary points of R_Ⅲ(n) but also proving the Schwa... 详细信息

关键词： Holomorphic mapping Schwarz lemma at the boundary The classical domain of typeⅢ

The Coefficient Inequalities for a Class of Holomorphic Mappings in Several Complex Variables

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Chinese Annals of Mathematics,Series B 2020年第1期41卷 37-48页

作者： Qinghua XU taishun liu Xiaosong liu School of Science Zhejiang University of Science and TechnologyHangzhou 310023China Department of Mathematics Huzhou Teacher’s UniversityHuzhou 313000ZhejiangChina School of Mathematics and Computation Science Lingnan Normal UniversityZhanjiang 524048GuangdongChina

The authors establish the coefficient inequalities for a class of holomorphic mappings on the unit ball in a complex Banach space or on the unit polydisk in C^n,which are natural extensions to higher dimensions of som... 详细信息

关键词： Coefficient inequality Fekete-Szego problem Quasi-convex mappings

Sharp distortion theorems for a subclass of close-to-convex mappings

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Frontiers of Mathematics in China 2013年第6期8卷 1425-1436页

作者： Qinghua XU taishun liu Xiaosong liu College of Mathematics and Information Science Jiangxi Normal University Nanchang 330022 China Department of Mathematics Huzhou Teachers College Huzhou 313000 China School of Mathematics and Computation Science Zhanjiang Normal University Zhanjiang 524048 China

我们介绍班强烈在一个复杂 Banach 空格的联合起来的球的顺序的 close-to-convex mappings，然后，我们在 n 在复杂 Hilbert 空间 X 或单位 polydisc 的联合起来的球为 mappings 的这个班给锋利的失真定理。作为应用，一条锋利的生长定... 详细信息

我们介绍班强烈在一个复杂 Banach 空格的联合起来的球的顺序的 close-to-convex mappings，然后，我们在 n 在复杂 Hilbert 空间 X 或单位 polydisc 的联合起来的球为 mappings 的这个班给锋利的失真定理。作为应用，一条锋利的生长定理为强烈，顺序的 close-to-convex mappings 被获得。

关键词：凹凸映射增长定理失真复Hilbert空间复Banach空间子类夏普单位球

Fekete and Szeg problem in one and higher dimensions

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Science China Mathematics 2018年第10期61卷 1775-1788页

作者： Qinghua Xu taishun liu Xiaosong liu School of Science Zhejiang University of Science and Technology Hangzhou 310023 China Department of Mathematics Huzhou University Huzhou 313000 China School of Mathematics and Computation Science Lingnan Normal University Zhanjiang 524048 China

In this paper, we establish the Fekete and Szeg 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 Banac... 详细信息

关键词：尺寸 Banach 函数 Cn

Rigidity for convex mappings of Reinhardt domains and its applications

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Science China Mathematics 2019年第5期62卷 901-920页

作者： taishun liu Xiaomin Tang Jianfei Wang Department of Mathematics Huzhou University School of Mathematical Sciences Huaqiao University Department of Mathematics Zhejiang Normal University

In this paper, we investigate rigidity and its applications to extreme points of biholomorphic convex mappings on Reinhardt domains. By introducing a version of the scaling method, we precisely construct many unbounded convex mappings with only one in?nite discontinuity on the boundary of this domain. We also give a rigidity of these unbounded convex mappings via the Kobayashi metric and the Liouville-type theorem of entire functions. As an application we obtain a collection of extreme points for the class of normalized convex mappings. Our results extend both the rigidity of convex mappings and related extreme points from the unit ball to Reinhardt domains.

关键词： convex mapping Reinhardt domain scaling method Kobayashi metric extreme point

Fekete and Szego problem for a subclass of quasi-convex mappings in several complex variables

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Frontiers of Mathematics in China 2015年第6期10卷 1461-1472页

作者： Qinghua XU Ting YANG taishun liu Huiming XU College of Mathematics and Information Science Jiangxi Normal University Nanchang 330022 China Department of Mathematics Huzhou University Huzhou 313000 China College of Mathematics Physics and Information Engineering Zhejiang Normal UniversityJinhua 321004 China

Let K be the familiar class of normalized convex functions in the unit disk. Keogh and Merkes proved the well-known result that maxf∈A ｜a3 - λa22｜ ≤ max{1/3, ｜λ - 1}, ,λ ∈ C, and the estimate is sharp for each ∈. We investigate the corresponding problem for a subclass of quasi-convex mappings of type B defined on the unit ball in a complex Banach space or on the unit polydisk in Cn. The proofs of these results use some restrictive assumptions, which in the case of one complex variable are automatically satisfied.

关键词： Fekete-Szego problem quasi-convex mappings of type A quasi-convex mappings of type B quasi-convex mappings of type C