准凸映照齐次展开式的精细估计

本文给出C~n中单位多圆柱上和复Banach空间中单位球上的准凸映照(含A型准凸映照和B型准凸映照)f齐次展开式的精细估计,其中x=0是f(x)-x的k 1阶零点.同时,还讨论了复Banach空间单位球上准凸映照的构造,它为准凸映照齐次展开式的精细估计提供极值映照.     

On quasi-convex mappings of orderαin the unit ball of a complex Banach space Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

In this paper, a class of biholomorphic mappings named quasi-convex mapping of order a 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 a defined on the polydisc in Cn and on the unit ball in a complex Banach space, respectively.     

多复变数一类α次星形映射齐次展开式各项的精细估计

本文首先给出复Banach空间单位球上一类α次星形映射齐次展开式各项的精细估计,特别当这些映射又是k折对称映射时,估计还是精确的.其次建立C~n中单位多圆柱上上述推广映射齐次展开式各项的精细估计,同样当这些映射又是k折对称映射时,估计仍是精确的.由此证明了多复变数中关于α次星形映射的弱Bieberbach猜想,且所得到的估计都能回到单复变数的情形.     

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.     

两类正规化双全纯映照子族齐次展开式的精细估计

本文考虑C~n中单位多圆柱上和一般复Banach空间中单位球上的正规化双全纯α(0■α＜1)次的殆β(-π/2＜β＜π/2)型螺形映照以及α(0＜α＜1)次的β(-π/2＜β＜π/2)型螺形映照f(其中x=0是f(x)-x的k+1阶零点),研究了它们的构造,并得到其齐次展开式的精细估计．所得的结果推广了以前相应的结论．     

Starlike and convex rational mappings on infinite dimensional domains

We give starlike criteria for a class of rational mappings on the open unit ball of a complex Banach space. We also give a sufficient condition for these mappings to be convex when they are defined in Hilbert spaces. These criteria facilitate the construction of concrete examples of starlike and convex mappings on infinite dimensional domains (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

一类近于凸映照子族精确的偏差定理

首先建立了C~n中单位多圆柱上一类近于凸映照子族精确的偏差定理,同时在复Banach空间单位球上也建立了该类映照精确的偏差定理的下界估计.其次在复Banach空间单位球上建立了准星形映照精确的偏差定理.所得结果将单复变中近于凸函数和星形函数的偏差定理推广至高维情形,并且对龚升提出的一个公开问题给出肯定的回答.     

α型螺形映照精细的增长、掩盖定理及精细的展开式系数估计

本文考虑一般复Banach空间中单位球上和Cn中单位多圆柱上的正规化双全纯α(-π/2<α<π/2)型螺形映照f(其中x=0是f(x)-x的k+1阶零点),研究了它的构造,并得到其精细的增长、掩盖定理以及齐次展开式的精细估计．所得的结果包含了许多已知的结论．     

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

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.     

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

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 ℂ ⁿ , 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 ℂ ⁿ and on the unit ball in a complex Banach space, respectively. Dedicated to Professor Sheng GONG on the occasion of his 75th birthday     

在Banach空间中推广的Roper-Suffridge算子(Ⅰ)

本文将Cn中的Roper-Suffridge算子推广到任意复Banach空间中,并证明这种算子在任意复Banach空间中的某些区域上具有保持ε星形性,由此可以构造出任意复Banach空间,复Hilbert空间和Cn中的一些区域上的许多双全纯星形映照、双全纯凸映照、双全纯ε星形映照,同时,得到它们的增长定理等,将龚升与刘太顺,Roper与suffridge,Graham,Kohr等学者在Cn中的一些结果推广到任意复Banach空间或复Hilbert空间中．     

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 ?ⁿ are also obtained. Especially, the above estimates are only sharp for a subclass of starlike mappings, quasi-convex mappings and quasi-convex mappings of type \(\mathbb{A}\). The results are the generalization of many known results.     

一类α次殆星形映照的偏差定理

近年来,双全纯凸(准凸)映照的偏差定理的研究已经获得一些可喜成果,但目前星形映照的偏差定理研究成果还较少.利用α次殆星形映射的增长估计,我们得到了C~n中开单位球B~n上一类α次殆星形映射的偏差估计.对复Banach空间单位球B和Reinhard域Ω_p1,…,p_n,可得到同样的结论,正如猜想的一样.     

一类多复变全纯映照子族的增长和偏差定理

在一般复Banach空间X中的单位球B上引入一类全纯映照族M_g.考虑B上满足条件(Df(x))~(-1)f(x)∈M_g的正规化局部双全纯映照f(x)(其中x=0是f(x)-x的k+1阶零点)并得到其增长定理.作为应用,也得到了C~n中单位多圆柱D~n上映照f关于Jacobi矩阵Jf(z)的偏差定理,该结果统一和推广了星形映照许多子族的相应结论.     

On Growth and Covering Theorem for Spiallike Mapping of Type β with Order α

In this paper,we consider growth and covering theorem for f(x),where f(x)is spiallike mapping of type β with order α defined on unit ball B of complex Banach space,and x=0 is zero of order k+1 for f(x)-x.We also dicate that the estimation is precise when β=0 and still give growth upper bound and distortion upper bound for subordinate mapping.This result include some results known.     

α次β型螺形映照的增长与掩盖定理

In this paper,we consider growth and covering theorem for f(x),where f(x) is spiallike mapping of typeβwith orderαdefined on unit ball B of complex Banach space, and x=0 is zero of order k+1 for f(x)-x.We also dicate that the estimation is precise whenβ=0 and still give growth upper bound and distortion upper hound for subordinate mapping.This result include some results known.     

相对于A的螺形映照的齐次展开式估计

运用k(k为自然数)阶零点的概念,给出了复Banach空间中相对于A的螺形映照f(x=0是f(x)-x的k+1阶零点)的齐次展开式的第k+1到2k项的估计结果.     

多复变数某些双全纯映射子族精确的系数估计

作者建立了复Banach空间单位球上和Cn中单位多圆柱上限制条件下双全纯映射齐次展开式的精确估计和 Fekete-Szeg?不等式,同时给出 Cn中 DP1,P2,…,Pn ={z ∈ Cn:n∑l=1|zl|Pl＜1}(pl＞1,l = 1,2,…,n)上限制条件下双全纯映射主要系数的精确估计和Fekete-Szeg...     

$\mathbb{C}^n$中单位球上$\rho$次抛物星形映射的一些估计

主要研究了$\mathbb{C}^n$中单位球$B^{n}$上$\rho \ (\rho \in [0,1))$次抛物星形映射的一些几何性质, 给出了该映射类的增长定理和掩盖定理, 及其齐次展开式中二次项的一些估计.     

Extension operators via semigroups

The Roper-Suffridge extension operator and its modifications are powerful tools to construct biholomorphic mappings with special geometric properties. The first purpose of this paper is to analyze common properties of different extension operators and to define an extension operator for biholomorphic mappings on the open unit ball of an arbitrary complex Banach space. The second purpose is to study extension operators for starlike, spirallike and convex in one direction mappings. In particular, we show that the extension of each spirallike mapping is A-spirallike for a variety of linear operators A. Our approach is based on a connection of special classes of biholomorphic mappings defined on the open unit ball of a complex Banach space with semigroups acting on this ball.