Loewner Chains and the Roper–Suffridge Extension Operator

Graham, Ian; Kohr, Gabriela; Kohr, Mirela

doi:10.1006/jmaa.2000.6843

Cited by 58 publications

(42 citation statements)

References 12 publications

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“…Let ( 1 ) ∈ * ( , , ) with −1 ≤ < < ( +1)/2 < 1 and ∈ (− /2, /2). Let ( , ) be the mapping denoted by (4) with…”

Section: The Invariance Of * ω ( )mentioning

confidence: 99%

“…Roper and Suffridge proved the Roper-Suffridge operator preserves convexity and star-likeness on . Graham et al generalized the RoperSuffridge operator and discussed the generalized operators preserving star-likeness and the block property in [4,5]. In 2002, Graham et al extended the Roper-Suffridge operator on the unit ball in C and proved the extended operator preserves star-likeness and convexity if and only if some conditions are satisfied in [6].…”

Section: Introductionmentioning

confidence: 99%

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Geometric Mappings under the Perturbed Extension Operators in Complex Systems Analysis

Wang

Cui

Liu

2017

Discrete Dynamics in Nature and Society

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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 type and order , * Ω ( , , ) as well as almost spiral-like mappings of type and order under different conditions on Bergman-Hartogs domains. Sequentially we obtain the conclusions on the unit ball and 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.

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“…Let ( 1 ) ∈ * ( , , ) with −1 ≤ < < ( +1)/2 < 1 and ∈ (− /2, /2). Let ( , ) be the mapping denoted by (4) with…”

Section: The Invariance Of * ω ( )mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Geometric Mappings under the Perturbed Extension Operators in Complex Systems Analysis

Wang

Cui

Liu

2017

Discrete Dynamics in Nature and Society

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“…[15] (see also ref. [10]), the authors proved that the Roper-Suffridge extension operator preserves the notion of parametric representation.…”

Section: Loewner Chains and The Roper-suffridge Extension Operatormentioning

confidence: 99%

The Roper-Suffridge extension operator and classes of biholomorphic mappings

Graham

Kohr

2006

SCI CHINA SER A

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In this paper we give a survey about the Roper-Suffridge extension operator and the developments in the theory of univalent mappings in several variables to which it has led. We begin with the basic geometric properties (most of which now have a number of different proofs) and discuss relations with the theory of Loewner chains and generalizations and modifications of the operator, some of which are very recent.

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“…Because RoperSuffridge extension operator has these important properties, many authors are interested in this extension operator. They generalized this extension operator in C n and discussed their properties (see [2,3,5,6,7,8,9,10], etc.). In [9], I.…”

Section: Introductionmentioning

confidence: 99%

“…They generalized this extension operator in C n and discussed their properties (see [2,3,5,6,7,8,9,10], etc.). In [9], I. Graham The purpose of the present paper is to extend the Roper-Suffridge extension operator from C n to Banach spaces and discuss its properties.…”

Section: Introductionmentioning

confidence: 99%

On the generalized Roper‐Suffridge extension operator in Banach spaces

Liu

Zhu

2005

International Journal of Mathematics and Mathematical Sciences

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The generalized Roper-Suffridge extension operator in Banach spaces is introduced. We prove that this operator preserves the starlikeness on some domains in Banach spaces and does not preserve convexity in some cases. Furthermore, the growth theorem and covering theorem of the corresponding mappings are given. Some results of Roper and Suffridge and Graham et al. in C n are extended to Banach spaces.

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Loewner Chains and the Roper–Suffridge Extension Operator

Cited by 58 publications

References 12 publications

Geometric Mappings under the Perturbed Extension Operators in Complex Systems Analysis

Geometric Mappings under the Perturbed Extension Operators in Complex Systems Analysis

The Roper-Suffridge extension operator and classes of biholomorphic mappings

On the generalized Roper‐Suffridge extension operator in Banach spaces

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