Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 41, 2016, 601-616
Kyushu Sangyo University, Faculty of Engineering
3-1 Matsukadai 2-Chome, Higashi-ku Fukuoka 813-8503, Japan;
h.hamada 'at' ip.kyusan-u.ac.jp
Babes-Bolyai University, Faculty of Mathematics and Computer Science
1 M. Kogalniceanu Str., 400084
Cluj-Napoca, Romania; miancu 'at' math.ubbcluj.ro
Babes-Bolyai University, Faculty of Mathematics and Computer Science
1 M. Kogalniceanu Str., 400084
Cluj-Napoca, Romania; gkohr 'at' math.ubbcluj.ro
Abstract. Let ŠAt(Bn) be the family of normalized univalent mappings on the Euclidean unit ball Bn in Cn, which have generalized parametric representation with respect to time-dependent operators A ∈ Ã, where à is a family of measurable mappings from [0,∞) into L(Cn) with some particular properties. Also, let RT(idBn,(NA(t))t∈[T0,T]) be the time-T-reachable family of the control system C([T0,T],(NA(t))t∈[T0,T]), where A ∈ à and T0 ≥ 0. In this paper we obtain certain convergence results for the families ŠAt(Bn) and RT(idBn,(NA(t))t∈[T0,T]) with respect to the Hausdorff metric ρ on H(Bn). These results may be seen as dominated convergence type theorems for time-dependent operators A ∈ Ã. In particular, we obtain related convergence results for the family SA0(Bn) (resp. for the family ŠA(Bn) of mappings with A-parametric representation on Bn (resp. of spirallike mappings on Bn with respect to A), in the case that A ∈ L(Cn) is a linear operator with k+(A) < 2m(A), where k+(A) is the Lyapunov index of A and m(A) = min||z||=1ℜ⟨A(z),z⟩. We also obtain a convergence result for the Carathéodory family NA, where m(A) > 0. Finally, we obtain some sufficient conditions related to A ∈ A, which yield the equality ŠAt(Bn) = S0(Bn), for all t ≥ 0, where S0(Bn) is the family of normalized univalent mappings with usual parametric representation on Bn.
2010 Mathematics Subject Classification: Primary 32H02; Secondary 30C45.
Key words: Carathéodory family, Hausdorff metric, Loewner chain, parametric representation, reachable family, transition mapping.
Reference to this article: H. Hamada, M. Iancu and G. Kohr: Convergence results for families of univalent mappings on the unit ball in Cn. Ann. Acad. Sci. Fenn. Math. 41 (2016), 601-616.
doi:10.5186/aasfm.2016.4136
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