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.

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 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 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).