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On the sharp growth covering theorems for normalized biholomorphic mappings in \(\mathbb{C}^n\). (English) Zbl 1150.30005

Summary: In this article, a normalized biholomorphic mapping \(f\) defined on bounded starlike circular domain in \(\mathbb{C}^n\) is considered, where \(z=0\) is a zero of order \(k+1\) of \(f (z)-z\). The sharp growth, covering theorems for almost starlike mappings of order \(\alpha\) and starlike mappings of order \(\alpha\) are established. Meanwhile, the construction of the above mappings on bounded starlike circular domain in \(\mathbb{C}^n\) is also discussed, it provides the extremal mappings for the growth, covering theorems of the above mappings.

MSC:

30C25 Covering theorems in conformal mapping theory
32A30 Other generalizations of function theory of one complex variable
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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