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 Research on Geometric Mappings in Complex Systems Analysis

2016 ◽  
Vol 2016 ◽  
pp. 1-11 ◽  
Author(s):  
Yanyan Cui ◽  
Chaojun Wang ◽  
Sifeng Zhu
Keyword(s):  
Banach Spaces ◽  
Systems Analysis ◽  
Starlike Functions ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Coefficient Estimates ◽  
Positive Real ◽  
Reinhardt Domains ◽  
Complex Order ◽  
Starlike Mappings

We mainly discuss the properties of a new subclass of starlike functions, namely, almost starlike functions of complex order λ, in one and several complex variables. We get the growth and distortion results for almost starlike functions of complex order λ. By the properties of functions with positive real parts and considering the zero of order k, we obtain the coefficient estimates for almost starlike functions of complex order λ on D. We also discuss the invariance of almost starlike mappings of complex order λ on Reinhardt domains and on the unit ball B in complex Banach spaces. The conclusions contain and generalize some known results.

scholarly journals On the Fekete and Szegö Problem for the Class of Starlike Mappings in Several Complex Variables

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Qing-Hua Xu ◽  
Tai-Shun Liu
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Unit Disk ◽  
Univalent Functions ◽  
Complex Banach Space ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Unit Polydisk ◽  
Starlike Mappings

LetSbe the familiar class of normalized univalent functions in the unit disk. Fekete and Szegö proved the well-known resultmaxf∈S⁡a3-λa22=1+2e-2λ/(1-λ)forλ∈0, 1. We investigate the corresponding problem for the class of starlike mappings defined on the unit ball in a complex Banach space or on the unit polydisk inCn, which satisfies a certain condition.

A theorem of continuation for functions of several complex variables

1958 ◽  
Vol 54 (3) ◽  
pp. 377-382 ◽  
Author(s):  
J. G. Taylor
Keyword(s):  
Field Theory ◽  
Circular Domain ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Quantum Field ◽  
Domain Of Holomorphy ◽  
Envelope Of Holomorphy ◽  
Special Cases ◽  
Theory Of Fields ◽  
Holomorphic Continuation

In the last few years it has been found useful to apply known theorems in the theory of functions of several complex variables to solve problems arising in the quantum theory of fields (11). In particular, in order to derive the dispersion relations of quantum field theory from the general postulates of that theory it appears useful to apply known theorems on holomorphic continuation for functions of several complex variables ((2), (10)). The most important theorems are those which enable a determination to be made of the largest domain to which every function which is holomorphic in a domain D may be continued. This domain is called the envelope of holomorphy of D, and denoted by E(D). If D = E(D) then D is termed a domain of holomorphy. E(D) may be defined as the smallest domain of holomorphy containing D. Only in the special cases that D is a tube, semi-tube, Hartogs, or circular domain has it been possible to determine the envelope of holomorphy E(D) ((3), (7)). An iterative method for the computation of envelopes of holomorphy has recently been given by Bremmerman(4). It is also possible to use the continuity theorem (1) in a direct manner, though in most cases this is exceedingly difficult.

scholarly journals A Note on Strongly Starlike Mappings in Several Complex Variables

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Hidetaka Hamada ◽  
Tatsuhiro Honda ◽  
Gabriela Kohr ◽  
Kwang Ho Shon
Keyword(s):  
Unit Ball ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Biholomorphic Mapping ◽  
Strongly Starlike ◽  
Starlike Mappings ◽  
Euclidean Unit Ball

Letfbe a normalized biholomorphic mapping on the Euclidean unit ball𝔹ninℂnand letα∈0,1. In this paper, we will show that iffis strongly starlike of orderαin the sense of Liczberski and Starkov, then it is also strongly starlike of orderαin the sense of Kohr and Liczberski. We also give an example which shows that the converse of the above result does not hold in dimensionn≥2.

Parametric Representation of Univalent Mappings in Several Complex Variables

2002 ◽  
Vol 54 (2) ◽  
pp. 324-351 ◽  
Author(s):  
Ian Graham ◽  
Hidetaka Hamada ◽  
Gabriela Kohr
Keyword(s):  
Unit Ball ◽  
Existence Theorems ◽  
Parametric Representation ◽  
Positive Real Part ◽  
Complex Variables ◽  
Holomorphic Mappings ◽  
Several Complex Variables ◽  
Positive Real ◽  
Loewner Equation ◽  
Euclidean Unit Ball

AbstractLet B be the unit ball of with respect to an arbitrary norm. We prove that the analog of the Carathéodory set, i.e. the set of normalized holomorphic mappings from B into of “positive real part”, is compact. This leads to improvements in the existence theorems for the Loewner differential equation in several complex variables. We investigate a subset of the normalized biholomorphic mappings of B which arises in the study of the Loewner equation, namely the set S0(B) of mappings which have parametric representation. For the case of the unit polydisc these mappings were studied by Poreda, and on the Euclidean unit ball they were studied by Kohr. As in Kohr’s work, we consider subsets of S0(B) obtained by placing restrictions on the mapping from the Carathéodory set which occurs in the Loewner equation. We obtain growth and covering theorems for these subsets of S0(B) as well as coefficient estimates, and consider various examples. Also we shall see that in higher dimensions there exist mappings in S(B) which can be imbedded in Loewner chains, but which do not have parametric representation.

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 An Expansion Theorem Involving H-Function of Several Complex Variables

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Sébastien Gaboury ◽  
Richard Tremblay
Keyword(s):  
Rational Function ◽  
Fractional Derivatives ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Expansion Theorem ◽  
General Expansion ◽  
Special Cases

The aim of this present paper is to obtain a general expansion theorem involving H-functions of several complex variables. This is done by making use of a Taylor-like expansion in terms of a rational function obtained by means of fractional derivatives given recently by the authors. Special cases are also computed.

Convex Subordination Chains in Several Complex Variables

2009 ◽  
Vol 61 (3) ◽  
pp. 566-582 ◽  
Author(s):  
Ian Graham ◽  
Hidetaka Hamada ◽  
Gabriela Kohr ◽  
John A. Pfaltzgraff
Keyword(s):  
Unit Ball ◽  
Sufficient Conditions ◽  
Complex Variables ◽  
Necessary And Sufficient Conditions ◽  
Several Complex Variables ◽  
Sufficient Condition ◽  
Subordination Chain ◽  
Euclidean Unit Ball ◽  
Necessary And Sufficient

Abstract.In this paper we study the notion of a convex subordination chain in several complex variables. We obtain certain necessary and sufficient conditions for a mapping to be a convex subordination chain, and we give various examples of convex subordination chains on the Euclidean unit ball in ℂn. We also obtain a sufficient condition for injectivity off(z/‖z‖, ‖z‖) onBn\ ﹛0﹜, wheref(z,t) is a convex subordination chain over (0, 1).

scholarly journals Universal functions in several complex variables

1979 ◽  
Vol 28 (2) ◽  
pp. 189-196 ◽  
Author(s):  
P. S. Chee
Keyword(s):  
Unit Ball ◽  
Universal Function ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Inner Function ◽  
Open Unit ◽  
Closed Unit Ball ◽  
Blaschke Products ◽  
Universal Functions ◽  
Open Unit Ball

AbstractIt is proved that there exists a universal good inner function in the open unit polydisc Un, that is its non Euclidean translates are dense in the closed unit ball of H∞ (Un) and that there exists a universal function in the open unit ball Bn of Cn. These generalize Heins' result on universal Blaschke products.1980 Mathematics subject classification (Amer. Math. Soc.): primary 32 A 10.

scholarly journals Kernel convergence and biholomorphic mappings in several complex variables

2003 ◽  
Vol 2003 (67) ◽  
pp. 4229-4239 ◽  
Author(s):  
Gabriela Kohr
Keyword(s):  
Unit Ball ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Compact Sets ◽  
Growth Theorem ◽  
Convex Mappings ◽  
Loewner Chains ◽  
Kernel Convergence

We deal with kernel convergence of domains inℂnwhich are biholomorphically equivalent to the unit ballB. We also prove that there is an equivalence between the convergence on compact sets of biholomorphic mappings onB, which satisfy a growth theorem, and the kernel convergence. Moreover, we obtain certain consequences of this equivalence in the study of Loewner chains and of starlike and convex mappings onB.

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