Convex subordination chains and injective mappings in \(\mathbb C^n\). (English) Zbl 1187.32012
Let \(\mathbb{C}^{n}\) be the space of \(n\) complex variables. Denote by \(B^{n}\) the open unit ball in \(\mathbb{C}^{n}\).
Let \(J\) be an interval in \(\mathbb{R}\). A mapping \(f=f(z,t):B^{n}\times J\rightarrow\mathbb{C}^{n}\) is said to be a convex subordination chain over \(J\) if the following conditions hold:
Let \(J\) be an interval in \(\mathbb{R}\). A mapping \(f=f(z,t):B^{n}\times J\rightarrow\mathbb{C}^{n}\) is said to be a convex subordination chain over \(J\) if the following conditions hold:
- (i)
- \(f(0,t)=0\) and \(f(.,t)\) is convex (biholomorphic) for \(t\in J\).
- (ii)
- \(f(.,t_{1})\prec f(.,t_{2})\) for \(t_{1},t_{2}\in J\;,\;t_{1}\leq t_{2}\).
In the paper, the authors obtain a sufficient criterion for \[ f(z,t)=a(t^{2})Df(tz)(tz)+f(tz)\;,\;z\in B^{n}\;,\;t\in[0,1] \] to be a convex subordination chain over \((0,1]\). The authors also find certain coefficient bounds which provide sufficient conditions for univalence, quasiregularity and starlikeness for the chain \(f(z,t)\). Some examples of convex subordination chains over \((0,1]\) are given.
Reviewer: Dorina Raducanu (Brasov)
MSC:
| 32H02 | Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables |
| 32A10 | Holomorphic functions of several complex variables |
Keywords:
biholomorphic mapping; convex mapping; convex subordination chain; Hadamard product; quasiconformal mapping; quasiregular mapping; starlike mapping; subordinationReferences:
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