Criteria for biholomorphic convex mappings on the unit ball in Hilbert spaces. (English) Zbl 1103.32008
Let \(X\) be a complex Hilbert space.The unit ball in \(X\) is \(B=\{z\in\text{X}:\| z\| <1\}.\) A biholomorphic mapping \(f:B\rightarrow X\) is called biholomorphic convex if \((1-t)f(z_1)+tf(z_2)\in f(B)\), for all \(z_1,z_2 \in B\) and \(0\leq t\leq1\).
The authors obtain a necessary and sufficient condition for a biholomorphic mapping \(f:B\rightarrow X\) to be biholomorphic convex, which improves some results due to H. Hamada and G. Kohr [Rev. Roum. Math. Pures Appl. 47, No. 3, 315–328 (2002; Zbl 1092.32501)]. Sufficient conditions for biholomorphic to be biholomorphic convex mappings and concrete examples of biholomorphic convex mappings are also given.
The authors obtain a necessary and sufficient condition for a biholomorphic mapping \(f:B\rightarrow X\) to be biholomorphic convex, which improves some results due to H. Hamada and G. Kohr [Rev. Roum. Math. Pures Appl. 47, No. 3, 315–328 (2002; Zbl 1092.32501)]. Sufficient conditions for biholomorphic to be biholomorphic convex mappings and concrete examples of biholomorphic convex mappings are also given.
Reviewer: Dorina Raducanu (Brasov)
MSC:
| 32H02 | Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables |
Keywords:
Biholomorphic convex mapping; Biholomorphic starlike mapping; Locally biholomorphic mappingCitations:
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