On Subordination Chains with Normalization Given by a Time-dependent Linear Operator

Abstract. Let S t A (B n ) be the family of normalized univalent mappings on the Euclidean unit ball B n in C n , which have generalized parametric representation with respect to time-dependent operators A ∈ A, where A is a family of measurable mappings from [0, ∞) into L(C n ) with some particular properties. Also, let, where A ∈ A and T 0 ≥ 0. In this paper we obtain certain convergence results for the families S t A (B n ) and) with respect to the Hausdorff metric ρ on H(B n ). These results may be seen as dominated convergence type theorems for time-dependent operators A ∈ A. In particular, we obtain related convergence results for the family S 0 A (B n ) (resp. for the family S A (B n )) of mappings with A-parametric representation on B n (resp. of spirallike mappings on B n with respect to A), in the case that A ∈ L(C n ) is a linear operator with k + (A) < 2m(A), where k + (A) is the Lyapunov index of A and m(A) = min z =1 ℜ A(z), z . We also obtain a convergence result for the Carathéodory family N A , where m(A) > 0. Finally, we obtain some sufficient conditions related to A ∈ A, which yield the equality S t A (B n ) = S 0 (B n ), for all t ≥ 0, where S 0 (B n ) is the family of normalized univalent mappings with usual parametric representation on B n . Certain consequences are also provided.

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“…Various properties of mappings with generalized parametric representation may be found in [12], [14], and [22].…”

Section: Preliminariesmentioning

confidence: 99%

Convergence results for families of univalent mappings on the unit ball in C^n

Hamada

Iancu

Kohr

2016

Ann. Acad. Sci. Fenn. Math.

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“…He generalized to higher dimensions the Loewner differential equation and developed existence and uniqueness theorems for its solutions. The existence and regularity theory (including changes in normalization such as those considered in this paper) has been considered by several authors, and applications have been given to the characterization of subclasses of biholomorphic mappings, univalence criteria, geometric characterizations of biholomorphic mappings with parametric representation (see [15], [20], [21], [23], [24], [25], [27], [29], [44], [45], [51]). A new approach to Loewner theory in the unit disc and complete hyperbolic complex manifolds, based on iteration and semigroup theory, may be found in [2], [3], [4], [7], [8], [9], [12], [34].…”

Section: Introductionmentioning

confidence: 99%

Univalent Subordination Chains in Reflexive Complex Banach Spaces

Graham¹,

Hamada²,

Kohr³

et al. 2013

Complex Analysis and Dynamical Systems V

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In this paper we consider univalent subordination chains in reflexive complex Banach spaces, allowing the chains to be normalized in terms of a positive linear operator. Related adaptations in the generalized Loewner differential equation and in the notion of parametric representation are also considered. The results in this paper are generalizations to reflexive complex Banach spaces of classical and recent results in the theory of Loewner chains and the Loewner differential equation on the unit ball in C n. Finally, we conclude with certain open problems and conjectures.

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“…For several results on subordination chains in several complex variables, the readers may consult [1,6,[10][11][12][13][14][15][16][19][20][21][24][25][26][28][29][30][31][32] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Polynomially bounded solutions to the Loewner differential equation in several complex variables

Hamada

2011

Journal of Mathematical Analysis and Applications

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We determine the form of polynomially bounded solutions to the Loewner differential equation that is satisfied by univalent subordination chains of the form f (z, t) = e t A z + · · · , where A ∈ L(C n , C n ) has the property m( A) > 0. Here m( A) = min{ A(z), z : z = 1}. We also give sufficient conditions for g(z, t) = L( f (z, t)) to be polynomially bounded, where f (z, t) is an A-normalized polynomially bounded Loewner chain solution to the Loewner differential equation.

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On Subordination Chains with Normalization Given by a Time-dependent Linear Operator

Cited by 10 publications

References 21 publications

Convergence results for families of univalent mappings on the unit ball in C^n

Convergence results for families of univalent mappings on the unit ball in C^n

Univalent Subordination Chains in Reflexive Complex Banach Spaces

Polynomially bounded solutions to the Loewner differential equation in several complex variables

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