On Spirallike Mappings in Several Complex Variables

In this paper,the parametric representation of a kind of spirallike mappings is given. As a direct application,the growth Theorem is set up. Moreover,a character of starlike mappings and a kind of spirallike mappings is obtained.

THE INVARIANCE OF STRONG AND ALMOST SPIRALLIKE MAPPINGS OF TYPE β AND ORDER α

The invariance of strong and almost spirallike mappings of type β and order α is discussed in this paper. From the maximum modulus principle of holomorphic functions, we obtain that the generalized Roper-Suffridge operators preserve strong and almost spirallike-hess of type β and order α on the unit ball B^n in C^n and on bounded and complete Reinhardt domains. Therefore we obtain that the generalized Roper-Suffridge operators preserve strong spirllikeness of type β, strong and almost starlikeness of order α, strong starlikeness on the corresponding domains.Thus we can construct more subclasses of spirallike mappings in several complex variables.

A Note of Spirallike Mappings in C^n

In this paper,the Growth and covering Theorems for a kind of spirallike mapping 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.

The Growth Theorem and Distortion Theorem of Spirallike Mappings on B_p

In this paper, by using the parametric representation of spirallike mappings, we construct to obtain the growth theorem of spirallike mappings on Reinhardt domain Bp. Moreover, the distortion theorem of spirallike mappings is obtained.

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.

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

In this paper, we deduce growth and covering theorem for f（x） by the other means,where f（x） is strongly spirallike mapping of type β with order α defined on Unit Ball B of complex Banach space, and still give growth upper bound and distortion upper bound for subordinate mapping.

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.

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.

The Growth and 1/4-Theorems for Spirallik Maps in l^P, B^P and Inner Product Space

The paper is made of two parts.In first part,We give the growth and 1/4-theorems for spiral like maps on the unit ball in l^p.Particularly,corresponding results were given in B^p.In the second part,we give the growth and 1/4-theorems for spirallike maps in an inner product space.We prove that the results is best.

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.

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 orderλ,almost spirallikeness of typeβand orderα,parabolic spirallikeness of typeβand orderρon the Hartogs domains in different conditions.They conclude that the corresponding extension operator preserves the same geometric invariance on the unit ball B^(n) in C^(n).The conclusions provide a new approach to study these geometric mappings in C^(n).

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

Let f（x） be an almost spirallike mapping of type β with order B on unit ball B of complex Banach space X. In this paper, we consider the growth and covering theorems for f（x）, we also prove 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.

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.

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

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.