Distortion theorems for normalized biholomorphic quasi-convex mappings

In this paper, we will use the Schwarz lemma at the boundary to character the distortion theorems of determinant at the extreme points and distortion theorems of matrix on the complex tangent space at the extreme points for normalized locally biholomorphic quasi-convex mappings in the unit ball B ⁿ respectively.     

The generalized Roper-Suffridge extension operator in Banach spaces (II)

In this paper, we generalize the Roper-Suffridge extension operator from Cn to Banach spaces. It is proved that this operator preserves the biholomorphic ? starlikeness on some domains in Banach spaces. From these, we may construct a lots of concrete examples about biholomorphic ? starlike mappings on some domains Ω in Cn, or Hilbert spaces, or Banach spaces from univalent ? starlike functions on the unit disc U in C. Meanwhile, the growth theorems of the corresponding mappings are given. Some results of Gong and Liu, Roper and Suffridge, Graham et al. in Cn are extended to Hilbert spaces or Banach spaces.     

On Biholomorphy in Infinite Dimensions

In this article, we consider the question of when a holomorphic mapping on a domain in a complex Hilbert space into itself has a holomorphic inverse. We give a strengthened version of a known result that involves a Fredholm condition on the mapping. We show that holomorphic mappings on certain domains that are biholomorphic near the boundary are biholomorphic on the domain itself.      

Criteria for biholomorphic convex mappings on the unit ball in Hilbert spaces

In this paper, we give a necessary and sufficient condition that a locally biholomorphic mapping f on the unit ball B in a complex Hilbert space X is a biholomorphic convex mapping, which improves some results of Hamada and Kohr and solves the problem which is posed by Graham and Kohr. From this, we derive some sufficient conditions for biholomorphic convex mapping. We also introduce a linear operator in purpose to construct some concrete examples of biholomorphic convex mappings on B in Hilbert spaces. Moreover, we give some examples of biholomorphic convex mappings on B in Hilbert spaces.     

On the roper-suffridge extension operator

In 1995, Roper and Suffridge defined an extension operator which maps a locally biholomorphic function on the unit diskD in ℂ to a locally biholomorphic mapping on the unit ballB ⁿ in ℂⁿ. This extension operator preserves many important properties, e.g., convexity and starlikeness, etc. In this note, we introduce the family ofε starlike mappings, and prove that the Roper-Suffridge extension operator preserves theε starlikeness on some Reinhardt domains. This result includes many known results and solves an open problem of Graham and Kohr. Project supported by the National Science Foundation of China.     

Finitely valent locally biholomorphic mappings of bounded domains

Considering countably connected domains of the plane, we solve the problem of existence of finitely valent locally biholomorphic mappings of a bounded domain onto a disk. We obtain sufficient conditions for the existence of the corresponding mappings in higher dimensions.     

On the Schwarz symmetry principle in a model case

In this article, we prove that smooth CR diffeomorphisms between two real analytic holomorphically nondegenerate hypersurfaces, one of which is rigid and polynomial, extend to be locally biholomorphic. It turns out that the result can be generalized to not totally degenerate mappings, in the sense of Baouendi and Rothschild.

     

多复变数双全纯映照的一些偏微分不等式

本文在有界平衡域上建立一些与双全纯映照有关的偏微分不等式．作为应用，给出 有界平衡域上星形映照以及近似星形映照的一些充分判别条件．     

The invariance of inner and outer linearly locally connected sets under quasiconformal mappings

In this paper, we prove that the inner and outer linearly locally connected sets of Rn are invariant respectively under quasiconformal mappings which fixed the point at infinity. On the other hand, we prove the converses of above results to be true also.     

The estimates of all homogeneous expansions for a subclass of biholomorphic mappings which have parametric representation in several complex variables

In this paper, we obtain the estimates of all homogeneous expansions for a subclass of biholomorphic 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 C~n. 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.     

Bloch Constant of Holomorphic Mappings on the Unit Ball of ℂ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>*

In this paper, the authors establish distortion theorems for various subfamilies Hk(B) of holomorphic mappings defined in the unit ball in Cn with critical points, where k is any positive integer. In particular, the distortion theorem for locally biholomorphic mappings is obtained when k tends to oo. These distortion theorems give lower bound son det f(z) and Redet 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.     

Hartogs域上延拓算子的性质

主要研究Roper-Suffridge延拓算子在推广的Hartogs域上的性质.借助双全纯映照的偏差定理,得到延拓算子在Ω_N上保持强α次殆β型螺形映照、α次殆β型螺形映照和α次β型螺形映照的性质,进而得到B~n上相应的结论.所得结论包含已有的结果并为研究C~n中的双全纯映照提供了新的途径.     

Applications holomorphes propres entre certains domaines bornés de ℂn

In this Note, we developp a new technic to study the existence of proper holomorphic mappings between a strictly pseudoconvex domain and certain special non-regular domains in ℂⁿ. In particular, it can be applied in the case of the minimal ball and the Lie ball. We prove that self-proper holomorphic mappings between such domains are biholomorphic. Furthermore, we establish a necessary and sufficient condition to factorize a proper holomorphic mapping by automorphisms. Scaling method is applied for the first time in a singular points of the boundary of the domains.     

DISTORTION THEOREM FOR BIHOLOMORPHIC MAPPINGS IN TRANSITIVE DOMAINS （Ⅳ）

The distortion theorem for biholomorphic starlike mappings (with respect to origin )in bounded symmetric domains are given,The distortion theorem for locally biholomorphic convex mappings in bounded symmetric domains are given also.     

Complete Quasiconvex Mappings in Polydisc

In this paper, a class of biholomorphic mappings called complete quasiconvex mappings is introduced and studied in bounded convex Reinhardt domains of ℂ ⁿ . Through a detailed analysis of the analytic characterization for this class of mappings, it is shown that this class of mappings contains the convex mappings and is also a subset of the class of starlike mappings. In the special case of the polydisc, a decomposition theorem is established for the complete quasiconvex mappings, which in turn is used to derive an improved sufficient condition for the convex mappings. Translated from Chinese Annals of Mathematics (Series A)     

On ε quasi-convex mappings in the unit ball of a complex Banach space

In this paper, a new class of biholomorphic mappings named “ε quasi-convex mapping” is introduced in the unit ball of a complex Banach space. Meanwhile, the definition of ε-starlike mapping is generalized from ε∈[0,1] to ε∈[−1,1]. It is proved that the class of ε quasi-convex mappings is a proper subset of the class of starlike mappings and contains the class of ε starlike mappings properly for some ε∈[−1,0)∪(0,1]. We give a geometric explanation for ε-starlike mapping with ε∈[−1,1] and prove that the generalized Roper-Suffridge extension operator preserves the biholomorphic ε starlikeness on some domains in Banach spaces for ε∈[−1,1]. We also give some concrete examples of ε quasi-convex mappings or ε starlike mappings for ε∈[−1,1] in Banach spaces or Cn. Furthermore, some other properties of ε quasi-convex mapping or ε-starlike mapping are obtained. These results generalize the related works of some authors.     

Extension Operators Preserving Biholomorphic Mappings on Hartogs Domains*

In this paper, the authors extend the Roper-Suffridge operator on the generalized Hartogs domains. They mainly research the properties of the extended operator.By the characteristics of Hartogs domains and the geometric properties of subclasses of spirallike mappings, they obtain the extended Roper-Suffridge operator preserving almost starlikeness of complex order λ, almost spirallikeness of type β and order α, parabolic spirallikeness of type β and order ρ on the Hartogs domains in different co...     

B~n上推广的Roper-Suffridge算子的性质

讨论改进后的Roper-Suffridge延拓算子保持双全纯映照子族的性质.借助双全纯映照子族的解析特征及其偏差结论,得到改进后的Roper-Suffridge延拓算子在一定条件下保持α次殆β型螺形映照、α次β型螺形映照、强β型螺形映照的性质,从而得到改进后的算子在一定条件下保持α次殆星形性、α次星形性和强星形性.所得结论为在多复变数空间中构造这些双全纯映照提供了一种新的途径.     

DISTORTION THEOREM FOR BIHOLOMORPHICMAPPINGS IN TRANSITIVE DOMAINS(IV)

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