Solutions for the generalized Loewner differential equation in several complex variables

Sci. China Ser. A-Math.

2010

Self Cite

In this paper we consider a proper subclassŜ n A of the full class of spirallike mappings on the Euclidean unit ball B n in C n with respect to a given linear operator A. We use the method of subordination chains to obtain an upper growth result forŜ n A , and we obtain various examples of mappings in the same class of normalized biholomorphic mappings on the unit ball B n in C n . We also prove that the classŜ n A is compact, while the full class of spirallike mappings with respect to a linear operator need not be compact in dimension n 2, even when the operator is diagonal. This is one of the motivations for considering the classŜ n A . Finally we prove that if f is a quasiregular strongly spirallike mapping on B n such that [Df (z)] −1 Af (z) is uniformly bounded on B n , then f extends to a homeomorphism of R 2n onto itself. In addition, if A + A * = 2aIn for some a > 0, this extension is also quasiconformal on R 2n .

Section: Definitionmentioning

confidence: 82%

Homeomorphic extension of strongly spirallike mappings in ℂ n

Curt

Sci. China Ser. A-Math.

2010

Self Cite

“…The above subordination implies the existence of the transition mapping v(z, s, t) associated with f (z, t), such that f (z, s) = f (v(z, s, t), t) for z ∈ B n and 0 ≤ s ≤ t < ∞. [6]). Also, in view of (2.2) and (2.3), we easily deduce that the condition (…”

Section: Definition 21 Letmentioning

confidence: 94%

“…The case A(t) ≡ I n was considered by Graham et al [15], and the case A(t) ≡ A ∈ L(C n , C n ) by Duren et al [6]. Note that if f (z, t) is the canonical solution of (2.7) and if g(z, t) = ( f (z, t)), where ∈ H (C n ) is such that (0) = 0, then g(z, t) is a standard solution of (2.7).…”

Section: Introductionmentioning

confidence: 99%

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

Graham

Hamada

Complex Anal. Oper. Theory

2010

Self Cite

In this paper we are concerned with solutions, in particular with univalent solutions, of the Loewner differential equation associated with non-normalized subordination chains on the Euclidean unit ball B n in C n . The main result is a generalization to higher dimensions of a well known result due to Becker. Various particular cases of this result have been recently obtained for subordination chains with normalization D f (0, t) = e t I n or D f (0, t) = e t A , t ≥ 0, where A ∈ L(C n , C n ). We also determine the form of the standard solutions to the Loewner differential equation associated with generalized spirallike mappings. In the last section we obtain the form of the solution in the presence of coefficient bounds. 788 I. Graham et al.

“…A mapping h : B n × [0, ∞) → C n which satisfies the conditions (i)-(iii) of Definition 2.4 will be called a Herglotz vector field (or a generating vector field) with respect to A (cf. [6] and [9]). …”

Section: Preliminariesmentioning

confidence: 98%

“…Since the early works devoted to Loewner chains and the Loewner differential equation in higher dimensions due to Pfaltzgraff [27] and Poreda [28,29], many results in this field have been obtained (see [1,5,6,9,11,13,14,15,20,21,35]). We also mention the main contributions of Bracci [5] related to the existence of bounded support points for the family S 0 (B n ), n ≥ 2, and of Roth [31] concerning the ndimensional version of the well known Pontryagin maximum principle.…”

Section: Introductionmentioning

confidence: 99%

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

Hamada

Iancu

Ann. Acad. Sci. Fenn. Math.

2016

Self Cite

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.