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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 *** 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

Support points for families of univalent mappings on bounded symmetric domains

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Science China Mathematics 2020年第12期63卷 2379-2398页

作者： Hidetaka Hamada Gabriela Kohr Faculty of Science and Engineering Kyushu Sangyo UniversityFukuoka 813-8503Japan Faculty of Mathematics and Computer Science Babecs-Bolyai UniversityCluj-Napoca 400084Romania

In this paper,we study some extremal problems for the family Sg^0(BX)of normalized univalent mappings with g-parametric representation on the unit ball BX of an n-dimensional JB*-triple X with r≥2,where r is the rank of X and g is a convex(univalent)function on the unit disc U,which satisfies some natural *** obtain sharp coefficient bounds for the family Sg^0(BX),and examples of bounded support points for various subsets of Sg^0(BX).Our results are generalizations to bounded symmetric domains of known recent results related to support points for families of univalent mappings on the Euclidean unit ball B^n and the unit polydisc U^n in C^*** questions will be also ***,we point out sharp coefficient bounds and bounded support points for the family Sg^0(B^n)and for special compact subsets of Sg^0(B^n),in the case n≥2.

关键词： bounded symmetric domain Carathéodory family g-Loewner chain parametric representation starlike mapping support point

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

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Science China Mathematics 2015年第12期58卷 2531-2540页

作者： LIU XiaoSong LIU TaiShun XU QingHua School of Mathematics and Computation Science Lingnan Normal University Department of Mathematics Huzhou Teachers College College of Mathematics and Information Science Jiangxi Normal University

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.

关键词： Bieberbach conjecture homogeneous expansion k-fold symmetric mapping a zero of order k + 1 starlike mapping

The Estimates of All Homogeneous Expansions for a Subclass of Biholomorphic mappings Which Have Parametric Representation in Several Complex Variables

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Acta Mathematica Sinica,English Series 2017年第2期33卷 287-300页

作者： Xiao Song LIU Tai Shun LIU School of Mathematics and Statistics Lingnan Normal University Department of Mathematics Huzhou Teachers College

In this paper, we obtain the estimates of all homogeneous expansions for a subclass of bi- holomorphic mappings which have parametric representation on the unit ball of complex Banach spaces. Meanwhile, we also establish the estimates of all homogeneous expansions for the above generalized mappings on the unit polydisk in Cn. Especially, the above estimates are only sharp for biholomorphic starlike mappings and starlike mappings of order α under restricted conditions. Our derived results generalize many known results.

关键词： Estimate of homogeneous expansion starlike mapping starlike mapping of order α quasi-convex mapping quasi-convex mapping of type A parametric representation

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： Homogeneous expansion ε quasi-convex mapping ε starlike mapping starlike mapping Quasi-convex mapping Quasi-convex mappings of type A

The Covering Theorems for Normalized Holomorphic mappings on Bounded starlike Circular Domains in C^n

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Chinese Quarterly Journal of Mathematics 2004年第1期19卷 1-5页

作者： LIUTai-shun FENGShu-xia DepartmentofMathematics UniversityofScienceandTechnologyofChinaHefei230026China

In this paper, we will investigate convex domains and starlike domains which contain the image set of normalized holomorphic mappings on bounded starlike circular domains in Cn.

关键词： covering theorem bounded starlike circular domain starlike mapping convex mapping

A Remark on the Generalized Roper-Suffridge Extension Operator for Almost Stalike mappings of Order α

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Chinese Quarterly Journal of Mathematics 2007年第1期22卷 22-28页

作者： LIU Xiao-song FENG Shu-xia School of Mathematics and Computation Science Zhanjiang Normal University Zhanjiang 524048 China College of Mathematics and Information Science Henan University Kaifeng 475001 China

Suppose f is an almost starlike function of order α on the unit disk D. In this paper, we will prove that Фn,β2,γ2,…βn,γn（f）（z）=（f（z1）,（f（z1）/z1）^β2（f＇（z1））^γ2 z2,…,（f（z1）/z1）^βn（f＇（z1））^γnzn）1 preserves almost starlikeness of order α on Ωn,p2,…,pn={z=（z1,z2,…,zn）＇∈Cn：∑^n j=1｜zj｜^pj〈1},where 0〈p1≤2,pj≥1,j=2,…,n,are real numbers.

关键词： almost starlike mapping of order α starlike mapping generalized Roper- Surffridge extension operator

The Study for Some Subclasses of Biholomorphic mappings by an Unified Method

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Chinese Quarterly Journal of Mathematics 2006年第2期21卷 166-175页

作者： XU Qing-hua LIU Tai-shun Department of Mathematics University of Science and Technology of China Hefei 230026 China Department of Mathematics Huzhou Teacher＇s College Huzhou 313000 China

Let Ω∈ C^n be a bounded starlike circular domain with 0 ∈ Ω. In this paper, we introduce a class of holomorphic mappings Mg on Ω. Let f（z） be a normalized locally biholomorphic mapping on Ω such that Jf^-1 （z） f（z） ∈Mg and z = 0 is the zero of order k＋1 of f（z） - z. We obtain the growth and covering theorems for f（z）. Especially, as corollaries, we unify and generalize many known results. Moreover, in view of proofs of corollaries, the essential relations among the subclasses of starlike mappings are shown.

关键词： growth and covering theorem starlike mapping subclasses of starlike mappings zero of order k + 1