THE GENERALIZED ROPER-SUFFRIDGE OPERATOR ON THE UNIT BALL IN COMPLEX BANACH AND HILBERT SPACES

In this paper, we extend the Roper-Suffridge extension operator in complex Ba- nach space, and prove that the extended Roper-Suffridge operators preserve the properties of the subclasses of spirallike mappings on the unit bM1 in complex Banach spaces. Thereby, we promote the conclusions to the cases in complex Hilbert spaces. The conclusions provide new approaches to construct these subclasses of spiraJlike mappings in several complex variables.

The Generalized Roper-suffridge Operator in Banach Spaces

In this paper, by the definition of spirallike mapping of type β and order α ,we discuss that the generalized Roper-Suffridge extension operator preserves spirallikeness of type β and order α in complex Banach spaces.

A Simple Proof of the Lower Bounds of Distortion Theorems on the Roper-Suffridge Operator

Liczberski-Starkov firstfound a lower bound for ｜｜D（f）｜｜ near the origin, where f（z）=（F（z1）,√F1（z1）z2,…,√F＇（z1）zn）＇is the Roper-Suffridge operator on the unit ball Bn in Cn and F is a normalized convex function on the unit disk. Later, Liczberski-Starkov and Hamada-Kohr proved the lower bound holds on the whole unit ball using a complex computation. Here we provide a rather short and easy proof for the lower bound. Similarly, when F is a normalized starlike function on the unit disk, a lower bound of ｜｜D（f）｜｜ is obtained again.

MODIFIED ROPER-SUFFRIDGE OPERATOR FOR SOME SUBCLASSES OF STARLIKE MAPPINGS ON REINHARDT DOMAINS

In this note, the author introduces some new subcIasses of starlike mappings S^＊Ωn1p2,…,pn（β,A,B）={f∈H（Ω）：｜itanβ＋（1-itanβ）2/p（z）аp/аz（z）Jf^-1（z）f（z）-1-AB/1-B^2｜〈B-A/1-B^2},on Reinhardt domains Ωn1p2,…,pn=z∈C^n：｜z1｜^2＋n∑j=2｜zj｜^pj〈1}where - 1≤A〈B〈1,q=min{p2,…,pn}≥1,l=max{p2,…,pn}≥2 and β ∈（-π/2,π/2）.Some different conditions for P are established such that these classes are preserved under the following modified Roper-Suffridge operator F（z）=（f（z1）＋f＇（z1）Pm（z0）,（f＇（z1））^1/mz0）＇where f is a normalized biholomorphic function on the unit disc D, z = （z1,z0） ∈Ωn1p2,…,pn,z0=（z2,…,zn）∈ C^n-1.Another condition for P is also obtained such that the above generalized Roper-Suffridge operator preserves an almost spirallike function of type/3 and order β These results generalize the modified Roper-Suffridge extension oper-ator from the unit ball to Reinhardt domains. Notice that when p2 = p3 …=pn = 2,our results reduce to the recent results of Feng and Yu.

The Spirallike Mapping of Type a and the Generalized Roper-Suffridge Operator

In this paper, we get the fact that generalized Roper-Suffridge operators act on spirallike mapping of type α on different domains is invariable, these results include some results known.

THE GENERALIZED ROPER-SUFFRIDGE EXTENSION OPERATOR

This note induces some generalized Roper-Suffridge extension operators such that they are used to construct some almost starlike mappings of order α and starlike mappings of order a on different domains.

PROPERTIES OF THE MODIFIED ROPER-SUFFRIDGE EXTENSION OPERATORS ON REINHARDT DOMAINS

In this paper, we mainly discuss the properties of the modified Roper-Suffridge operators on Reinhardt domains. By the analytical characteristics and distortion results of subclasses of biholomorphic mappings, we conclude that the modified Roper-Suffridge operators preserve the properties of SΩ （β, A, B）, almost starlike mapping of complex order A on Ωn,p2……pn. Sequentially, we get that the modified Roper-Suffridge operators preserve spirallikeness of type β and order a, strongly pirallikeness of type β and order a, almost starlikeness of order a on Ωn,p2……pn The conclusions provide a new approach to construct these biholomorphic mappings which have special geometric properties in several complex variables.

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 Generalized Roper-Suffridge Extension Operator for Locally Biholomorphic Mappings

In this paper, we introduce the generalized R oper-Suffridge extension operator for locally biholomorphic mappings. It is sh own that this operator preserves the starlikeness on some Reinhardt domains and does not preserve convexity for some cases. Meanwhile, the growth theorem and di stortion theorem of the corresponding mappings are given.

A Remark on the Generalized Roper-Suffridge Extension Operator for Spirallike Mappings of Type β and Order α

Suppose f is a spirallike function of type β and order α on the unit disk D.Let Ωn,p1,p2,…,pn={z=（z2,z2,…,zn）′∈C^n：∑j=1^n｜zj）^Pj〈1},where 1≤p1≤2,pj≥1,j=2,…,n,are real numbers.In this paper,we will prove that Φn,β2,γ2,…βn,γn（f）（z）=（f（z1）, preserves spirallikeness of type β and order α on Ωn,p1,p2,…,Pn.

One Roper-Suffridge Extension Operator in Complex Banach Spaces

In this paper, by the definition of almost spirallike mappings of type β and order α and the geometric property of the spirallike mapping of type β, we prove that the generalized Roper-Suffridge extension operator preserves almost spirallikeness of type β and order α in complex Banach spaces. Key words：

The Generalized Roper-Suffridge Extension Operator on the Reinhardt Domain

Let pj ∈ N and pj ≥-1, j = 2,...,n be a fixed positive integer. In this paper a generalized Roper-Suffridge extension operator F（z） ={f（Z1）＋f＇（z1）} on Reinhardt domain is defined. Some different conditions for Pj areestablished under which the operator preserves an almost spirallike mapping of type fl and order a and spirallike mapping of type β and order α, respectively. In particular, our results reduce to many well-known results.

Loewner Chains Associated with the Generalized Roper-Suffridge Extension Operator on the Unit Ball in Hilbert Spaces

In this paper, we prove that the generalized Roper-Suffridge extension operator can be embeded in Loewner chains on the unit ball in Hibert spaces, and obtain the fact that the operator keeps the properties of almost spirallike mapping of typeβ and order α, almost starlikeness of order α, spirallikeness of type ofβ and starlikeness.

Loewner Chains Associated with the Generalized Roper-Suffridge Extension Operator

In this paper, we prove that the generalized Roper-Suffridge extension operator can be embedded in Loewner chains on the unit ball in Hibert spaces, and obtain the fact that the operator keeps the properties of almost spirallike mapping of type β and other α, almost staxlikeness of order α, spirallikeness of type β and starlikeness.

CONVEX MAPPINGS ASSOCIATED WITH THE ROPER-SUFFRIDGE EXTENSION OPERATOR

Let λG（z）|dz| be the hyperbolic metric on a simply connected proper domain G?C containing the origin, and let ■be the Banach norms of Cnj for j = 1, 2,…,k.This note is to prove that if f is a normalized biholomorphic convex function on G, then ■is a normalized biholomorphic convex mapping on ■where N = 1+n1 + … + nk and the branch is chosen such that ■,j = 1,…, k. Applying to the Roper-Suffridge extension operator, we obtain a new convex mappings construction of an unbounded domain and a refinement of convex mappings construction on a Reinhardt domain, respectively.

A Remark on the Generalized Roper-Suffridge Extension Operator for Almost Stalike Mappings of Order α

Suppose f is an almost starlike function of order α on the unit disk D. In this paper, we will prove that Фn,β2,γ2,…βn,γn（f）（z）=（f（z1）,（f（z1）/z1）^β2（f＇（z1））^γ2 z2,…,（f（z1）/z1）^βn（f＇（z1））^γnzn）1 preserves almost starlikeness of order α on Ωn,p2,…,pn={z=（z1,z2,…,zn）＇∈Cn：∑^n j=1｜zj｜^pj〈1},where 0〈p1≤2,pj≥1,j=2,…,n,are real numbers.

New Subclasses of Biholomorphic Mappings and the Modified Roper-Suffridge Operator

The authors propose a new approach to construct subclasses of biholomorphic mappings with special geometric properties in several complex variables. The RoperSuffridge operator on the unit ball B^n in C^n is modified. By the analytical characteristics and the growth theorems of subclasses of spirallike mappings, it is proved that the modified Roper-Suffridge operator [Φ_(G,γ)(f)](z) preserves the properties of S_Ω~*(A, B), as well as strong and almost spirallikeness of type β and order α on B^n. Thus, the mappings in S_Ω~*(A, B), as well as strong and almost spirallike mappings, can be constructed through the corresponding functions in one complex variable. The conclusions follow some special cases and contain the elementary results.

全纯映射子族上改进的Roper-Suffridge算子

设m∈N且m≥2,f是单位圆盘D上的正规化双全纯函数,P:C^(n-1)→C是m次齐次多项式.研究了单位球上改进的Roper-Suffridge算子F(z)=(f(z_1)+f′(z_1)P(z_0),[f′(z_1)]^(1/m)z_0)′,其中z_0=(z_2,z_3,…,z_n).当m=2时,该算子由Muir引入.当P≡0和m=2时,此算子就是著名的Roper-Suffridge算子.给出了‖P‖分别满足不同条件时,改进的算子分别保持α次的殆星和α次的星形映射性质.此结果推广了最近许多已有结果,而且方法更易于操作.特别地,当f(z_1)=z_1和m=n=2时,这就导出了多复变几何函数论中构造具体实例的经典形式F(z)=(z_1+az_2~2,z_2)′.

两类推广的Roper-Suffridge算子的性质

讨论两类推广的Roper-Suffridge算子保持双全纯映照子族的性质.从定义出发证明推广的Roper-Suffridge算子在Cn中单位球上和复Banach空间单位球上保持强α次殆星映照及α次准凸映照的性质,进而得到推广的Roper-Suffridge算子在Cn中单位球上和复Hilbert空间单位球上分别保持强星形性和α次准凸性.