scholarly journals Some properties of Reinhardt and Hartogs domains

2002 ◽  
Vol 24 (24) ◽  
pp. 07
Author(s):  
Ludmila Bourchtein ◽  
Andrei Bourchtein
Keyword(s):  
Power Series ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Geometric Representation ◽  
Reinhardt Domain ◽  
Hartogs Domains ◽  
Domain Conservation

The domains of certain types, such as Reinhardt and Hartogs, are used in different problems of theory of functions of several complex variables. For instance, any power series of several complex variables converges in the complete logarithmically convex Reinhardt domain. Transformations of Reinhardt and Hartogs allow us to diminish the dimension of space and to consider some type of domains using its images. This allows the visibility of the geometric representation and the simplification of the study of properties of such domains. In this article we consider some properties of Reinhardt and Hartogs domains and transformations. The properties of domain conservation under Reinhardt transformation and convexity conservation under Reinhardt and Hartogs transformation are proved.

scholarly journals Meromorphic or Holomorphic Completion of a Reinhardt Domain

1970 ◽  
Vol 38 ◽  
pp. 1-12 ◽  
Author(s):  
Eiichi Sakai
Keyword(s):  
Meromorphic Function ◽  
Power Series ◽  
Meromorphic Functions ◽  
Integral Formula ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Reinhardt Domain ◽  
Continuity Theorem ◽  
Cauchy’S Integral Formula ◽  
Long Time

In the theory of functions of several complex variables, the problem about the continuation of meromorphic functions has not been much investigated for a long time in spite of its importance except the deeper result of the continuity theorem due to E. E. Levi [4] and H. Kneser [3], The difficulty of its investigation is based on the following reasons: we can not use the tools of not only Cauchy’s integral formula but also the power series and there are indetermination points for the meromorphic function of many variables different from one variable. Therefore we shall also follow the Levi and Kneser’s method and seek for the aspect of meromorphic completion of a Reinhardt domain in Cn.

scholarly journals Logarithmically Convex Reinhardt Domains

2003 ◽  
Vol 25 (25) ◽  
pp. 07 ◽  
Author(s):  
Ludmila Bourchtein ◽  
Andrei Bourchtein
Keyword(s):  
Power Series ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Reinhardt Domain ◽  
Reinhardt Domains ◽  
Logarithmic Convexity

The domains of certain types, such as Reinhardt ones, are important in different problems of theory of functions of several complex variables. For instance, any power series of several complex variables converges in the complete logarithmically convex Reinhardt domain. In this article we prove the logarithmic convexity of complete convex Reinhardt domain.

scholarly journals Geometric Mappings under the Perturbed Extension Operators in Complex Systems Analysis

2017 ◽  
Vol 2017 ◽  
pp. 1-14
Author(s):  
Chaojun Wang ◽  
Yanyan Cui ◽  
Hao Liu
Keyword(s):  
Complex Systems ◽  
Unit Ball ◽  
Systems Analysis ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Geometric Properties ◽  
Geometric Characteristics ◽  
New Approaches ◽  
Special Cases ◽  
Hartogs Domains

In this paper, we mainly seek conditions on which the geometric properties of subclasses of biholomorphic mappings remain unchanged under the perturbed Roper-Suffridge extension operators. Firstly we generalize the Roper-Suffridge operator on Bergman-Hartogs domains. Secondly, applying the analytical characteristics and growth results of subclasses of biholomorphic mappings, we conclude that the generalized Roper-Suffridge operators preserve the geometric properties of strong and almost spiral-like mappings of typeβand orderα,SΩ⁎(β,A,B)as well as almost spiral-like mappings of typeβand orderαunder different conditions on Bergman-Hartogs domains. Sequentially we obtain the conclusions on the unit ballBnand for some special cases. The conclusions include and promote some known results and provide new approaches to construct biholomorphic mappings which have special geometric characteristics in several complex variables.

scholarly journals Some Results of Fekete-Szegö Type. Results for Some Holomorphic Functions of Several Complex Variables

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1707
Author(s):  
Renata Długosz ◽  
Piotr Liczberski
Keyword(s):  
Power Series ◽  
Power Series Expansion ◽  
Holomorphic Functions ◽  
Positive Real Part ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Homogeneous Polynomials ◽  
Positive Real ◽  
Sharp Estimates ◽  
Several Variables

This paper is devoted to a generalization of the well-known Fekete-Szegö type coefficients problem for holomorphic functions of a complex variable onto holomorphic functions of several variables. The considerations concern three families of such functions f, which are bounded, having positive real part and which Temljakov transform Lf has positive real part, respectively. The main result arise some sharp estimates of the Minkowski balance of a combination of 2-homogeneous and the square of 1-homogeneous polynomials occurred in power series expansion of functions from aforementioned families.

scholarly journals A study on bornological properties of the space of entire functions of several complex variables

2002 ◽  
Vol 33 (4) ◽  
pp. 289-302 ◽  
Author(s):  
Mushtaq Shaker Abdul-Hussein ◽  
G. S. Srivastava
Keyword(s):  
Power Series ◽  
Vector Space ◽  
Entire Functions ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Locally Convex Spaces ◽  
Classical Analysis ◽  
Analysis Theory ◽  
Important Position ◽  
Locally Convex

Spaces of entire functions of several complex variables occupy an important position in view of their vast applications in various branches of mathematics, for instance, the classical analysis, theory of approximation, theory of topological bases etc. With an idea of correlating entire functions with certain aspects in the theory of basis in locally convex spaces, we have investigated in this paper the bornological aspects of the space $X$ of integral functions of several complex variables. By $Y$ we denote the space of all power series with positive radius of convergence at the origin. We introduce bornologies on $X$ and $Y$ and prove that $Y$ is a convex bornological vector space which is the completion of the convex bornological vector space $X$.

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