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

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 X.We 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 X.The results that we derive include some results in several complex variables,and extend the classical result in one complex variable to several complex variables.

双曲度量和Roper-Suffridge算子

本文以双曲度量为工具,获得了复平面上单连通区域的若干分析性质,并且应用这些性质分别研究Roper-Suffridge算子和扰动的Roper-Suffridge算子的分析与几何特性.本文不仅给出利用双曲度量研究多复变数双全纯映照的新方法,而且所得结果更为一般和深刻.

多复变数全纯映射精细的Bohr定理

本文首先建立不依赖自同构从复Banach空间平衡域到Cn单位多圆柱上一定限制条件下全纯映射精细的范数型Bohr定理及复Banach空间X上单位球到复Banach空间Y上单位球全纯映射精细的泛函型Bohr定理.其次,给出有界对称域上全纯映射精细的Bohr定理.最后,得到J*代数单位球上全纯映射精细的Bohr定理.所得结果将一维的Bohr定理推广至高维.

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

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.

Bloch Constant of Holomorphic Mappings on the Unit Ball of C^n

In this paper, the authors establish distortion theorems for various subfamilies Hk（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 subfamilies βk（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 biholomorphic mappings is also obtained.

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

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.

Fekete and Szeg problem in one and higher dimensions

In this paper, we establish the Fekete and Szego 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 Cn.

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

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.

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

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).

Rigidity for convex mappings of Reinhardt domains and its applications

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.

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

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.

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

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.

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

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<1}p>1.In particular,the above distortion theorems are sharp if Bpn is the unit polydisk in Cn.Our results reduce to the corresponding classical results in one dimension of complex function theory.

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

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 some Fekete and Szego inequalities for subclasses of the normalized univalent functions in the unit disk.

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

We introduce the class of strongly close-to-convex mappings of order c~ in the unit ball of a complex Banach space, and then, we give the sharp distortion theorems for this class of mappings in the unit ball of a complex Hilbert space X or the unit polydisc in Cn. As an application, a sharp growth theorem for strongly close-to-convex mappings of order α is obtained.