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Sharp distortion theorems for some subclasses of star like mappings on Bnp in Cn

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Frontiers of Mathematics in China 2020年第1期15卷 127-140页

作者： xiaosong liu Taishun liu School of Mathematics and Statistics Lingnan Normal UniversityZhanjiang 524048China Department of Mathematics Huzhou UniversityHuzhou 313000China

We mainly establish the distortion theorems of Jacobi determinant for three subclasses of starlike mappings on Bpn,where Bp={z=(z1,……,zn)T∈Cn:∑nl=1|zl|p*** particular,the above distortion theorems are sharp if Bpn is the unit polydisk in *** results reduce to the corresponding classical results in one dimension of complex function theory.

关键词： Starlike mapping distortion theorem.Jacobi determinant

A SUBCLASS OF QUASI-CONVEX MAPPINGS ON A REINHARDT DOMAIN IN C^n

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Acta Mathematica Scientia 2020年第6期40卷 1709-1722页

作者： xiaosong liu School of Mathematics and Statistics Lingnan Normal UniversityZhanjiang 524048China

Let D p 1,p 2,⋯,p n={z∈C n:∑l=1 n|z l|p l1,l=1,2,⋯,*** this article,we first establish the sharp estimates of the main coefficients for a subclass of quasi-convex mappings(including quasi-convex mappings of type A and quasi-convex mappings of type B)on D p 1,p 2,⋯,p n under some weak additional ***,we also establish the sharp distortion theorems for the above *** results that we obtain reduce to the corresponding classical results in one dimension.

关键词： quasi-convex mapping quasi-convex mapping of type A quasi-convex mapping of type B main coefficient distortion theorem

SHARP DISTORTION THEOREMS FOR A CLASS OF BIHOLOMORPHIC MAPPINGS IN SEVERAL COMPLEX VARIABLES

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Acta Mathematica Scientia 2022年第2期42卷 454-466页

作者： xiaosong liu School of Mathematics and Statistics Lingnan Normal UniversityZhanjiang 524048China

In this paper,we first establish the sharp growth theorem and the distortion theorem of the Frechet derivative for biholomorphic mappings defined on the unit ball of complex Banach spaces and the unit polydisk in C^(n) with some restricted *** next give the distortion theorem of the Jacobi determinant for biholomorphic mappings defined on the unit ball of C^(n) with an arbitrary norm and the unit polydisk in C^(n) under certain restricted *** we obtain the sharp Goluzin type distortion theorem for biholomorphic mappings defined on the unit ball of complex Banach spaces and the unit polydisk in C^(n) with some additional *** results derived all reduce to the corresponding classical results in one complex variable,and include some known results from the prior literature.

关键词： Biholomorphic mapping distortion theorem Frechet derivative Jacobi determinant Goluzin type distortion theorem

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

The Sharp Estimates of all Homogeneous Expansions for a Class of Quasi-convex Mappings on the Unit Polydisk in C^n

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Chinese Annals of Mathematics,Series B 2011年第2期32卷 241-252页

作者： xiaosong liu Taishun liu School of Mathematics and Computation Science Zhanjiang Normal University Zhanjiang 524048 Guangdong China Department of Mathematics Huzhou Teachers College Huzhou 313000 Zhejiang China

In this paper,the sharp estimates of all homogeneous expansions for f are established,where f(z) = (f1(z),f2(z),··· ,fn(z)) is a k-fold symmetric quasi-convex mapping defined on the unit polydisk in Cn and Dtk+1fp(0)(ztk+1) (tk + 1)! = n l1,l2,···,ltk+1=1 |apl1l2···ltk+1|ei θpl1+θpl2+···+θpltk+1t k+1 zl1zl2 ··· zltk+1,p = 1,2,··· ,*** i = √?1,θplq ∈ (-π,π] (q = 1,2,··· ,tk + 1),l1,l2,··· ,ltk+1 = 1,2,··· ,n,t = 1,2,···.Moreover,as corollaries,the sharp upper bounds of growth theorem and distortion theorem for a k-fold symmetric quasi-convex mapping are established as *** results show that in the case of quasi-convex mappings,Bieberbach conjecture in several complex variables is partly proved,and many known results are generalized.

关键词： Estimates of all homogeneous expansions Quasi-convex mapping Quasi-convex mapping of type A Quasi-convex mapping of type B

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

A refinement about estimation of expansion coefficients for normalized biholomorphic mappings

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Science China Mathematics 2005年第7期48卷 865-879页

作者： liu Taishun liu xiaosong Department of Mathematics University of Science and Technology of China Hefei 230026 China Department of Mathematics University of Science and Technology of China Hefei 230026 China

A refining estimation of homogeneous expansion for / is discussed, where f belongs to a subclass of all normalized biholomorphic mappings defined on the unit polydisk in Cn or the unit ball in complex Banach spaces, and x = 0 is a zero of order k+1 of f(x)-x. Moreover, an estimation of homogeneous expansion for subordinate mappings defined on the unit ball in complex Banach spaces is also given.

关键词： de Branges theorem refining estimation of homogeneous expansion almost starlike mapping of order a starlike mapping of order a subordinate mapping.

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

A REFINEMENT OF THE SCHWARZ-PICK ESTIMATES AND THE CARATHéODORY METRIC IN SEVERAL COMPLEX VARIABLES

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Acta Mathematica Scientia 2024年第4期44卷 1337-1346页

作者： xiaosong liu Taishun liu School of Mathematics and Statistics Lingnan Normal UniversityZhanjiang 524048China Department of Mathematics Huzhou UniversityHuzhou 313000China

In this article,we first establish an asymptotically sharp result on the higher order Fréchet derivatives for bounded holomorphic mappings f(x)=f(0)+∞∑s=1Dskf(0)(x^(sk))/(sk)!:B_(X)→B_(Y),where B_X is the unit ball of *** next give a sharp result on the first order Fréchet derivative for bounded holomorphic mappings F(X)=F(0+)∞∑s=KD^(s)f(0)(x^(8)/s!):B_(X)→B_(Y),where B_(X)is the unit ball of *** results that we derive include some results in several complex variables,and extend the classical result in one complex variable to several complex variables.

关键词： refined Schwarz-Pick estimate bounded holomorphic mapping Carathéodory metric first order Fréchet derivative higher order Fréchet derivatives