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 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.

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.

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 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.

Holomorphic Functions with Positive Real Part

1982 ◽  
Vol 34 (1) ◽  
pp. 1-7 ◽  
Author(s):  
Eric Sawyer
Keyword(s):  
Unit Ball ◽  
Hardy Spaces ◽  
Convex Cone ◽  
Holomorphic Functions ◽  
Positive Real Part ◽  
Positive Real ◽  
Spaces Of Holomorphic Functions ◽  
Extreme Element ◽  
Image Position ◽  
Special Case

The main purpose of this note is to prove a special case of the following conjecture.Conjecture. If F is holomorphic on the unit ball Bn in Cn and has positive real part, then F is in Hp(Bn) for 0 < p < ½(n + 1).Here Hp(Bn) (0 < p < ∞) denote the usual Hardy spaces of holomorphic functions on Bn. See below for definitions. We remark that the conjecture is known for 0 < p < 1 and that some evidence for it already exists in the literature; for example [1, Theorems 3.11 and 3.15] where it is shown that a particular extreme element of the convex cone of functionsis in Hp(B2) for 0 < p < 3/2.

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.

scholarly journals Holomorphic mappings into the complex projective space with moving hypersurfaces

Filomat ◽  
2021 ◽  
Vol 35 (3) ◽  
pp. 955-962
Author(s):  
Liu Yang
Keyword(s):  
Projective Space ◽  
Recent Work ◽  
General Position ◽  
Complex Projective Space ◽  
Type Theorem ◽  
Complex Variables ◽  
Holomorphic Mappings ◽  
Several Complex Variables ◽  
Picard Type Theorem ◽  
Complex Projective

Motivated by Eremenko?s accomplisshment of a Picard-type theorem [Period Math Hung. 38 (1999), pp.39-42.], we study the normality of families of holomorphic mappings of several complex variables into PN(C) for moving hypersurfaces located in general position. Our results generalize and complete previous results in this area, especially the works of Dufresnoy, Tu-Li, Tu-Cao, Yang-Fang-Pang and the recent work of Ye-Shi-Pang.

scholarly journals Normal Families of meromorphic mappings of several complex variables into PN(C)

2005 ◽  
Vol 180 ◽  
pp. 91-110 ◽  
Author(s):  
Pham Ngoc Mai ◽  
Do Duc Thai ◽  
Pham Nguyen Thu Trang
Keyword(s):  
Complex Space ◽  
Sufficient Conditions ◽  
Complex Variables ◽  
Holomorphic Mappings ◽  
Several Complex Variables ◽  
Normal Families ◽  
Meromorphic Mappings ◽  
Compact Complex Space

AbstractThe first aim in this article is to give some sufficient conditions for a family of meromorphic mappings of a domain D in Cn into PN(C) omitting hypersurfaces to be meromorphically normal. Our result is a generalization of the results of Fujimoto and Tu. The second aim is to investigate extending holomorphic mappings into the compact complex space from the viewpoint of the theory of meromorphically normal families of meromorphic mappings.

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