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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 Frechet-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 Cn 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~ are obtained, and the corresponding results of the above generalized mappings on the unit polydisk in C~ are got as well. Thus, some known results in prior literatures are generalized.

关键词： Distortion theorem A zero of order k + 1 Frechet-derivative Jacobideterminant parametric representation

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

One Kind of Mappings with parametric representation in Cn

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Chinese Quarterly Journal of Mathematics 2014年第3期29卷 373-380页

作者： DENG Wei-bing LIU Hao School of Mathematics and Information Science Henan University Kaifeng 475001 China

In this paper, a kind of mappings with parametric representation is introduced,the growth theorem is set up.

关键词： parametric representation growth theorem

The parametric representation for Starlike Mappings on Bounded Starlike Circular Domains

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Chinese Quarterly Journal of Mathematics 2016年第3期31卷 288-297页

作者：闫春燕张芳芳 Department of Information Engineering Xuchang Vocational and Technical College College of Information Engineering Huanghe Science and Technology College

In this paper,the parametric representation of starlike mappings is *** a direct application,the growth theorem is set ***,a character of starlike mappings is obtained.

关键词： starlike mappings parametric representation growth theorem

parametric representationS OF QUASICONFORMAL MAPPINGS

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Acta Mathematica Scientia 2020年第6期40卷 1874-1882页

作者： Zhenlian LIN Qingtian SHI School of Mathematical Sciences Huaqiao UniversityQuanzhou 362021China School of Mathematics and Computer Science Quanzhou Normal UniversityQuanzhou 362000China

In this article,we first give two simple examples to illustrate that two types of parametric representation of the family ofΣ0 K have some *** we also find that the area derivative formula(1.6),which is used to estimate the area distortion ofΣ0 K,cannot be derived from[6],but that formula still holds forΣ0 K through our amendatory parametric representation for the one obtained by Eremenko and Hamilton.

关键词： quasiconformal mapping Σ0 K parametric representation area distortion theorem Cauchy transformation

The Growth Theorem and Distortion Theorem of Spirallike Mappings on B_p

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Acta Mathematica Sinica,English Series 2015年第5期31卷 811-824页

作者： Hao LIU Ke Ping LU Fang Fang ZHANG School of Mathematics and Statistics Institute of Contemporary MathematicsHe'nan University College of Mathematics and Information Sciences Xinyang Normal University School of Mathematics and Statistics He'nan University

In this paper, by using the parametric representation of spirallike mappings, we construct to obtain the growth theorem of spirallike mappings on Reinhardt domain Bp. Moreover, the distortion theorem of spirallike map... 详细信息

关键词： parametric representation spirallike mappings Reinhardt domains

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

Extreme points, support points and the Loewner variation in several complex variables

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Science China Mathematics 2012年第7期55卷 1353-1366页

作者： GRAHAM Ian HAMADA Hidetaka KOHR Gabriela KOHR Mirela Department of Mathematics University of Toronto Toronto Ontario M5S 2E4 Canada Faculty of Engineering Kyushu Sangyo University 3-1 Matsukadai 2-Chome Higashi-ku Fukuoka 8138503 Japan Faculty of Mathematics and Computer Science Babe s-Bolyai University 1 M. Koga lniceanu Str. Cluj-Napoca 400084 Romania Faculty of Mathematics and Computer Science Babes-Bolyai University 1 M. Kogǎlniceanu Str. Cluj-Napoca 400084 Romania

In this paper we consider extreme points and support points for compact subclasses of normalized biholomorphic mappings of the Euclidean unit ball Bn in Cn. We consider the class So（Bn） of biholomorphic mappings on Bn which have parametric representation, i.e., they are the initial elements f（-, O） of a Loewner chain f（x,t） = etz ＋ ... such that {e-tf（.,t）}t≥o is a normal family on Bn. We show that if f（.,O） is an extreme point （respectively a support point） of So（Bn）, then e-t f（., t） is an extreme point of So（Bn） for t≥0 （respectively a support point of So（Bn） for t C [O, t0] and some to〉 0）. This is a generalization to the n-dimensional case of work due to Pell. Also, we prove analogous results for mappings which belong to So（Bn） and which are bounded in the norm by a fixed constant. We relate the study of this class to reachable sets in control theory generalizing work of Roth. Finally we consider extreme points and support points for biholomorphic mappings of Bn generated by using extension operators that preserve Loewner chains.

关键词： biholomorphic mapping Loewner chain parametric representation reachable set starlike map-ping

On Spirallike Mappings in Several Complex Variables

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Chinese Quarterly Journal of Mathematics 1999年第1期14卷 62-72,页

作者：刘浩中国科学技术大学安徽合肥 2 3 0 0 2

In this paper,the parametric representation of a kind of spirallike mappings is given. As a direct application,the growth Theorem is set up. Moreover,a character of starlike mappings and a kind of spirallike mappings ... 详细信息

关键词： spirallike mappings parametric representation growth theorem