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| 1 | THE GROWTH THEOREM FOR STARLIKE MAPPINGS ON BOUNDED STARLIKE CIRCULAR DOMAINS显示文摘The authors obtain the growth and covering theorem for the class of normalized biholomorphic starlike mappings on bounded starlike circular domains.This type of domain discussed is rather general, since the domain must be starlike if there exists a normalized biholomorphic starlike mapping on it. In the unit disc, it is just the famous growth and covering theorem for univalent functions.This theorem successfully realizes the initial idea of H. Cartan about how to extend geometric function theory from one variable to several complex variables. | LIU TAISHUN (Department of Mathematics, University of Science and Technology of China, Hefei 230026, China)REN GUANGBIN (Department of Mathematics, University of Science and Technology of China, Hefei 230026, China) | 1998 | Chinese Annals of Mathematics,Series B1998，19，4： | 39 |
| 2 | A refinement about estimation of expansion coefficients for normalized biholomorphic mappings显示文摘A refining estimation of homogeneous expansion for f 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. | LIU Taishun LIU Xiaosong | 2005 | Science China Mathematics2005，48，7： | 18 |
| 3 | The Sharp Estimates of all Homogeneous Expansions for a Class of Quasi-convex Mappings on the Unit Polydisk in C^n显示文摘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,··· ,n.Here 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 well.These 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. | Xiaosong LIU Taishun LIU | 2011 | Chinese Annals of Mathematics,Series B2011，32，2： | 11 |
| 4 | On quasi-convex mappings of order α in the unit ball of a complex Banach space显示文摘In this paper, a class of biholomorphic mappings named quasi-convex mapping of order α in the unit ball of a complex Banach space is introduced. When the Banach space is confined to Cn, we obtain the relation between this class of mappings and the convex mappings.Furthermore, the growth and covering theorems of this class of mappings are given on the unit ball of a complex Banach space Ⅹ. Finally, we get the second order terms coefficient estimations of the homogeneous expansion of quasi-convex mapping of order α defined on the polydisc in Cn and on the unit ball in a complex Banach space, respectively. | LIU Taishun & XU Qinghua Department of Mathematics, Huzhou Teachers College, Huzhou, 313000, China Department of Mathematics, Jiangxi Normal University, Nanchang, 330022, China | 2006 | Science China Mathematics2006，49，11： | 9 |
| 5 | Bloch Constant of Holomorphic Mappings on the Unit Ball of C^n显示文摘In this paper,the authors establish distortion theorems for various subfamilies H_k(B)of holomorphic mappings defined in the unit ball in C^n with critical points,where k is any positive integer.In particular,the distortion theorem for locally biholomorphic mappings is obtained when k tends to+∞.These distortion theorems give lower bounds on|det f′(z)|and Re det f′(z).As an application of these distortion theorems,the authors give lower and upper bounds of Bloch constants for the subfamiliesβ_k(M)of holomorphic mappings.Moreover,these distortion theorems are sharp.When B is the unit disk in C,these theorems reduce to the results of Liu and Minda.A new distortion result of Re det f′(z)for locally biholomorphie mappings is also obtained. | Jianfei WANG Taishun LIU | 2007 | Chinese Annals of Mathematics,Series B2007，28，6： | 7 |
| 6 | A proof of a weak version of the Bieberbach conjecture in several complex variables显示文摘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. | LIU XiaoSong LIU TaiShun XU QingHua | 2015 | Science China Mathematics2015，58，12： | 7 |
| 7 | Schwarz lemma at the boundary of the unit polydisk in C^n显示文摘We establish a new type of the classical boundary Schwarz lemma for holomorphic self-mappings of the unit polydisk Dnin Cn. By using the Carath′eodory metric and Kobayashi metric of Dn, we obtain some properties of the complex Jacobian matrix Jf(p) at a boundary point p of Dnfor a holomorphic self-mapping f of Dn. Our results extend the classical Schwarz lemma at the boundary to high dimensions. | TANG XiaoMin LIU TaiShun LU Jin | 2015 | Science China Mathematics2015，58，8： | 6 |
| 8 | Fekete and Szego problem for a subclass of quasi-convex mappings in several complex variables显示文摘让 K 是在联合起来的磁盘的规范的凸的函数的熟悉的类。Keogh 和 Merkes 证明了著名结果是那最大 ? fK|a3−λa22|≤max ?{ 1/3 ,|λ−1|},λ? ,并且估计为每 λ
是锋利的。我们为在一个复杂 Banach 空格或在联合起来的 polydisk 上在联合起来的球定义在的类型 B 的伪凸的 mappings 的一个子类调查相应问题 ? n。这些结果的证明使用一些限制假设,在一建筑群的情况中,变量它自动地满足。 | Qinghua XU Ting YANG Taishun LIU Huiming XU | 2015 | Frontiers of Mathematics in China2015，10，6： | 5 |
| 9 | Fekete and Szeg problem in one and higher dimensions显示文摘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 Banach space or on the unit polydisk in C^n. | Qinghua Xu Taishun Liu Xiaosong Liu | 2018 | Science China Mathematics2018，61，10： | 3 |
| 10 | Bloch constant of holomorphic mappings on the unit polydisk of C^n显示文摘In this paper,we give a definition of Bloch mappings defined in the unit polydisk Dn, which generalizes the concept of Bloch functions defined in the unit disk D.It is known that Bloch theorem fails unless we have some restrictive assumption on holomorphic mappings in several complex variables.We shall establish the corresponding distortion theorems for subfamiliesβ(K)andβloc(K) of Bloch mappings defined in the polydisk Dn,which extend the distortion theorems of Liu and Minda to higher dimensions.As an application,we obtain lower and upper bounds of Bloch constants for various subfamilies of Bloeh mappings defined in Dn.In particular,our results reduce to the classical results of Ahlfors and Landau when n=1. | WANG JianFei~(1+) LIU TaiShun~2 1 College of Mathematics and Physics Information Engineering Zhejiang Normal University Jinhua 321004,China 2 Department of Mathematics Huzhou Teachers College Huzhou 313000 China | 2008 | Science China Mathematics2008，51，4： | 3 |
| 11 | Sharp Distortion Theorems for a Subclass of Biholomorphic Mappings Which Have a Parametric Representation in Several Complex Variables显示文摘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. | Xiaosong LIU Taishun LIU | 2016 | Chinese Annals of Mathematics,Series B2016，37，4： | 2 |
| 12 | Growth theorem of convex mappings on bounded convex circular domains显示文摘 | Taishun Liu Guangbin Ren | 1998 | Science in China Series A: Mathematics1998，，2： | 1 |
| 13 | On growth and covering theorems of quasi-convex mappings in the unit ball of a complex banach space显示文摘 | Zhang Wenjun Liu Taishun | 2002 | Science in China Series A: Mathematics2002，，12： | 1 |
| 14 | On Decomposition Theorem of normalized biholomorphic convex mappings in Reinhardt domains显示文摘 | Wenjun Zhang Taishun Liu | 2003 | Science in China Series A: Mathematics2003，，1： | 1 |
| 15 | Homogeneous expansions of normalized biholomorphic convex mappings overB P显示文摘 | Taishun Liu Wenjun Zhang | 1997 | Science in China Series A: Mathematics1997，，8： | 1 |
| 16 | Schwarz Lemma at the Boundary on the Classical Domain of Type Ⅲ显示文摘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 Schwarz lemma at the smooth boundary point for holomorphic self-mappings of R_Ⅲ(n). | Taishun LIU Xiaomin TANG Wenjun ZHANG | 2020 | Chinese Annals of Mathematics,Series B2020，41，3： | 1 |
| 17 | A boundary Schwarz lemma on the classical domain of type I显示文摘Let R_I(m,n) be the classical domain of type I in C^(m×n)with 1≤m≤n.We obtain the optimal estimates of the eigenvalues of the Fréchet derivative Df(Z) at a smooth boundary fixed point Z of R_I(m,n)for a holomorphic self-mapping f of R_L(m,n).We provide a necessary and sufficient condition such that the boundary points of R_I(m,n) are smooth,and give some properties of the smooth boundary points of R_L(m,n).Our results extend the classical Schwarz lemma at the boundary of the unit disk △ to R_I(m,n),which may be applied to get some optimal estimates in several complex variables. | LIU TaiShun TANG XiaoMin | 2017 | Science China Mathematics2017，60，7： | 1 |
| 18 | Homogeneous expansions of normalized biholomorphic convex mappings overB P显示文摘 | Taishun Liu Wenjun Zhang | 1997 | Science in China Series A: Mathematics1997，，8： | 1 |
| 19 | On Decomposition Theorem of normalized biholomorphic convex mappings in Reinhardt domains显示文摘 | Wenjun Zhang Taishun Liu | 2003 | Science in China Series A: Mathematics2003，，1： | 1 |
| 20 | On the roper-suffridge extension operator显示文摘 | Sheng Gong Taishun Liu | 2002 | Journal d’Analyse Mathématique2002，，1： | 1 |
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