On the Holomorphic Solutions of Certain Differential Equations of First Order for the Mappings of the Unit Ball in C" Into C"

Poreda, Tadeusz; Szadkowska, Anna

doi:10.1515/dema-1989-0406

Cited by 4 publications

(7 citation statements)

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“…In view of Corollary 4.6, we obtain the following existence and uniqueness result for solutions of the differential equation (4.2), which is related to spirallikeness on the unit ball in C n (see [49,Theorem 3] and [48,Theorem 4.3], in the case A < 2m(A), and [13,Proposition 3.7.5], in the case k + (A) < 2k − (A)).…”

Section: Corollary 44 Let a ∈ L(c N C N ) Be Such That K + (A) < 2mentioning

confidence: 96%

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Solutions for the generalized Loewner differential equation in several complex variables

et al. 2009

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We generalize a one-variable result of J. Becker to several complex variables. We determine the form of arbitrary solutions of the Loewner differential equation that is satisfied by univalent subordination chains of the form(The notion of parametric representation has a useful generalization under these conditions, so that one has a canonical solution of the Loewner differential equation.) In particular, we determine the form of the univalent solutions. The results are applied to subordination chains generated by spirallike mappings on the unit ball in C n . Finally, we determine the form of the solutions in the presence of certain coefficient bounds. I.

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Section: Corollary 44 Let a ∈ L(c N C N ) Be Such That K + (A) < 2mentioning

confidence: 96%

“…The next result refers to the case where h(·, t) ∈ N is independent of t (compare [48,Theorem 4.3], [49], in the case A < 2m(A), and [13,Proposition 3.7.5], [50], in the case k …”

Section: Corollary 44 Let a ∈ L(c N C N ) Be Such That K + (A) < 2mentioning

confidence: 99%

Solutions for the generalized Loewner differential equation in several complex variables

et al. 2009

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“…Poreda and Szadkowska [30], Lemma 1 and Theorem 3, proved the following result, in the case A < 2m(A) (see also [12,Proposition 3.7.5]). It was recently improved by Graham, Hamada, Kohr and Kohr [16], in the case k + (A) < 2m(A).…”

Section: Definitionmentioning

confidence: 85%

“…Clearly, if f ∈Ŝ n A , then f is also spirallike with respect to A by Lemma 3, and thus biholomorphic on We next prove the main result of this section (compare [30,Theorem 4]). We mention that in higher dimensions, there is no growth result for the full class of spirallike mappings (see [21,14,15]).…”

Section: Definition 11 Letmentioning

confidence: 90%

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Homeomorphic extension of strongly spirallike mappings in ℂ n

Curt

Kohr

2010

Sci. China Ser. A-Math.

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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 .

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Asymptotically spirallike mappings in several complex variables

et al. 2008

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In this paper, we define the notion of asymptotic spirallikeness (a generalization of asymptotic starlikeness) in the Euclidean space C n . We consider the connection between this notion and univalent subordination chains. We introduce the notions of A-asymptotic spirallikeness and A-parametric represen-, then a mapping f ∈ S(B n ) is Aasymptotically spirallike if and only if f has A-parametric representation, i.e., if and only if there exists a univalent subordination chain f (z, t) such that Df (0, t) = e At , {e −At f (·, t)} t≥0 is a normal family on B n and f = f (·, 0). In particular, a spirallike mapping with respect to A ∈ L(C n , C n ) with ∞ 0 e (A−2m(A)In )t dt < ∞ has A-parametric representation. We also prove that if f is a spirallike mapping with respect to an operator A such that A + A * = 2In, then f has parametric representation (i.e., with respect to the identity). Finally, we obtain some examples of asymptotically spirallike mappings.

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On the Holomorphic Solutions of Certain Differential Equations of First Order for the Mappings of the Unit Ball in C" Into C"

Cited by 4 publications

References 0 publications

Solutions for the generalized Loewner differential equation in several complex variables

Solutions for the generalized Loewner differential equation in several complex variables

Homeomorphic extension of strongly spirallike mappings in ℂ n

Asymptotically spirallike mappings in several complex variables

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