A NEW ROPER-SUFFRIDGE EXTENSION OPERATOR ON BOUNDED COMPLETE REINHARDT DOMAINS

In this paper, we construct a new Roper-Suffridge extension operator Φn^r,β1,,βn（f）（z） = F（z） = （（rf（z1/r）/z1）^β1z1,（rf（z1/r）/z1）^β2z2,...,（rf（z1/r）/z1）^βnzn）＇,where f is a normalized locally biholomorphic function on the unit disc D, r = sup{｜z1｜ ： z =（z1, ···, zn） ∈Ω}, β1∈ [0, 1], 0 ≤βk≤β1, k = 2, ···, n, then we prove it can preserve the property of spirallikeness of type β, almost starlikeness of order α and starlikeness of orderα on bounded complete Reinhardt domain Ω, respectively.

ROPER-SUFFRIDGE EXTENSION OPERATOR ON A REINHARDT DOMAIN

Let pj ∈ N and pj≥ 1, j = 2, …, k, k ≥ 2 be a fixed positive integer. We introduce a Roper-Suffridge extension operator on the following Reinhardt domain ΩN ={z =（z1, z′2,…, z′k）′∈ C × C^n2×…× Cnk： ｜z1｜^2＋ ｜｜z2｜｜2^p2＋ … ＋ ｜｜zk ｜｜k^pk〈 1} given〈1} give by F（z）=（f（z1）＋f＇（z1）∑j=2 kPj（zj,（f＇（z1））1/p2 z2＇,…,（f＇（z1））1/pkzk＇）＇, where f is a normaljized biholomorphic function k（z） =（f（z1） ＋ f′（z1）=2 on the unit disc D, and for 2 ≤ j ≤ k, Pj ： C^nj→ C is a homogeneous polynomial of degree pj and zj =（zj1, …, zjnj）′∈ C^nj, nj ≥ 1, pj ≥ 1,｜｜zj｜｜j =（∑l=1 nj｜zjl｜^pj）1/pj. In this paper, some conditions for Pjare found under which the loperator p ｜zjl｜pj=1reserves the properties of almost starlikeness of order α, starlikeness of order αand strongly starlikeness of order α on ΩN, respectively.

THE EXTENSION OPERATORS ON Bn+1 AND BOUNDED COMPLETE REINHARDT DOMAINS

In this article, we extend the well-known Roper-Suffridge operator on Bn+1 and bounded complete Reinhardt domains in Cn+1, then we investigate the properties of the generalized operators. Applying the Loewner theory, we obtain the mappings constructed by the generalized operators that have parametric representation on Bn+1. In addition, by using the geometric characteristics and the parametric representation of subclasses of spirallike mappings, we conclude that the extended operators preserve the geometric properties of several subclasses of spirallike mappings on Bn+1 and bounded complete Reinhardt domains in Cn+1. The conclusions provide new approaches to construct mappings with special geometric properties in Cn+1.

CLASSIFICATION OF A CLASS OF HYPERBOLIC REINHARDT DOMAINS

In this article, the author discusses the dimension of holomorphic automorphism groups on hyperbolic Reirihardt domains. and classifies those hyperbolic Reinhardt domains whose automorphism group has prescribed dimension n2 - 2 (where n is the dimension of domain).

Function Theory of a Class of Bounded Reinhardt Domains

在这份报纸，我们考虑围住的 Reinhardt 领域 D 的一个班(m， n1，， nm ) 。Bergman 核功能 K (z，(z)) ， Bergman 公制的矩阵 T (z，(z)) ， Cauchy-Szeg (o) 核功能 S (z，()) 被获得。然后，我们证明正式泊松内核函数不是一个泊松内核函数。最后，我们证明 D 是一个 quasiconvex 领域，如果并且仅当 D 是一个超球面， D 是一个更强壮的 quasiconvex 领域。

Convex Mappings on Some Reinhardt Domains

In this paper, we consider the following Reinhardt domains. Let M = （M1, M2,..., Mn） ： [0,1] → [0,1]^n be a C2-function and Mj（0） = 0, Mj（1） = 1, Mj″ 〉 0, C1jr^pj-1 〈 Mj′（r） 〈 C2jr^pj-1, r∈ （0, 1）, pj 〉 2, 1 ≤ j ≤ n, 0 〈 C1j 〈 C2j be constants. Define DM={z=（z1,z2,…,Zn）^T∈C^n：n∑j=1 Mj（｜zj｜）〈1}Then DM C^n is a convex Reinhardt domain. We give an extension theorem for a normalized biholomorphic convex mapping f ： DM -→ C^n.

Forward (Δ) and Backward (∇) Difference Operators Basic Sets of Polynomials in and Their Effectiveness in Reinhardt and Hyperelliptic Domains

We generate, from a given basic set of polynomials in several complex variables , new basic sets of polynomials and generated by the application of the Δ and &#8711 operators to the set . All relevant properties relating to the effectiveness in Reinhardt and hyperelliptic domains of these new sets are properly deduced. The case of classical orthogonal polynomials is investigated in details and the results are given in a table. Notations are also provided at the end of a table.

On Decomposition Theorem of normalized biholomorphic convex mappings in Reinhardt domains

The construction of normalized biholomorphic convex mappings in the Reinhardt domain Dp{(z1,z2,....zn):z1 p1+z1 p2+....+zn pn <1},(pj>2,j=1,2,.....n,) of n is discussed.The authors set up a Decomposition Theorem for such mappings. As a special case, it is proved that, for each suchthis derives the Decomposition Theorem of normalized biholo morphic convex mappings in the polydisc whichwas gotten by T.J. Suffridge in 1970.

The generalized Roper-Suffridge extension operator on bounded complete Reinhardt domains

The generalized Roper-Suffridge extension operator Ф(f) on the bounded complete Reinhardt domain Ω in Cn with n ≥ 2 is defined by Φrn,β2,γ2,…,βn,γn(f)(z)=(rf(z1/r),(rf(z1/r)/z1)β2(f'(z1/r))γ2z2,…,(rf(z1/r)/z1)βn(f'(z1/r)γnzn) for (z1,z2,…,zn) ∈Ω, where r = r(Ω) = sup{|z1|: (z1,z2,…,zn) ∈ Ω},0 ≤ γj ≤ 1 -βj,0 ≤ βj ≤ 1,and we choose the branch of the power functions such that (f(z1)/z1)βj |z1=0 = 1 and (f′(z1))γj |z1=0 =1,j = 2,…,n. In this paper, we prove that the operator Фrn,β2,γ2,…,βn,γn(f) is from the subset of S*α(U) to S*α(Ω)(0 ≤ α ＜ 1) on Ω and the operator Фrn,β2,γ2,…, βn,γn(f) preserves the starlikeness of order α or the spirallikeness of type β on Dp for some suitable constantsβj,γj,pj, where Dp ={(z1,z2,…,zn) ∈ Cn: ∑nj=1|zj|pj ＜ 1} (pj ＞ 0, j = 1,2,…,n), U is the unit disc in the complex plane C, and Sα* (Ω) is the class of all normalized starlike mappings of order α on Ω. We also obtain that Φrn,β2,γ2,…,γn(f) ∈ S*α(Dp) if and only if f ∈ S*a(U) for 0 ≤ α ＜ 1 and some suitable constants βj,γj,pj.

CURVATURES ON A CLASS OF REINHARDT DOMAINS

For a class of Reinhardt domains, we prove that the holomorphic sectional curvatures are upper-bounded by a negative constant. Then we obtain a comparison theorem for the Kobayashi and Bergman metrics on the domains.

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.

Reinhardt domains (Ⅱ)

The Bergman kernel function K(z,), z, w∈Ω for a domain ΩC^n is the kernel of the Bergman projection operator, the operator projecting L^2(Ω) onto its holomorphic subspace. In this note, we consider the Reinhardt

A NECESSARY AND SUFFICIENT CONDITION:BIHOLOMORPHIC MAPPINGS ARE STARLIKE ON A CLASS OF REINHARDT DOMAINS

1. In [1], T. J. Suffridge gave the necessary and sufficient condition for local biholomorphic mappings on the Reinhardt domains , to be starlike. Using this condition Yu Qihuang et al. gave the growth theorem and covering theorem for normalized biholomorphic starlike mappings in D_p. In this note we give a necessary and suffident condition for local biholomorphic mappings on more general Reinhardt

The Bergman kernel function of some Reinhardt domains (Ⅱ)

The boundary behavior of the Bergman kernel function of a kind of Reinhardt domain is studied. Upper and lower bounds for the Bergman kernel function are found at the diagonal points ( Z, Z) Let Q be the Reinhardt domainwhere is the Standard Euclidean norm in and let K( Z, W) be the Bergman kernel function of Ω. Then there exist two positive constants m and M, and a function F such thatholds for every Z∈Ω . Hereand is the defining function of Ω The constants m and M depend only on Ω = This result extends some previous known results.

Proper holomorphic self-maps of smooth bounded Reinhardt domains in C^2

It is proved that every proper holomorphic self-map of a smooth bounded Reinhardt domain in C^2 is an automorphism.

Proper holomorphic self-maps of smooth bounded Reinhardt domains in C^n

It is proved that every proper holomorphic self-map of a smooth bounded Reinhardt domain of D'Angelo finite type in Cn (n ＞ 1) is an automorphism.

一类华罗庚域的Bergman核函数的显式表达

在任意不可约有界圆型齐性域上考虑一类华罗庚域E(q 1,…,q m,Ω;p 1,…,p m),其中Ω是指任意不可约有界圆型齐性域,q 1,…,q m都是自然数,m,p 1,…,p m都是正整数,N(Z,Z)是Ω的一般范数。利用完备正交函数系和多元极坐标变换,给出了该域的Bergman核函数的显式表达。

Reinhardt域上推广的Roper-Suffridge算子与Loewner链

研究了Roper-Suffridge算子与Loewner链的关系,利用Loewner链的定义及性质证明了推广的Roper-Suffridge算子在有界完全Reinhardt域上能嵌入Loewner链,从α次殆β型螺形映照与Loewner链的关系出发得到结论:推广后的Roper-Suffridge算子在有界完全Reinhardt域上保持α次殆β型螺形性,并作为特殊情况得出推广后的Roper-Suffridge算子在相应域上保持α次殆星形性及β型螺形性.

Reinhardt域上和复Hilbert空间单位球上推广的Roper-Suffridge算子的性质

由α次的殆β型螺形映照的定义,分别给出推广的Roper-Suffridge算子在Reinhardt域上和复Hilbert空间中的单位球上保持α次的殆β型螺形性.