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Growth and Distortion Results for a Class of Biholomorphic Mapping and Extremal Problem with Parametric Representation in \(\mathbb {C}^n\)

  • Zhenhan Tu
  • Liangpeng XiongEmail author
Article
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Abstract

Let \(\widehat{\mathcal {S}}_g^{\alpha , \beta }(\mathbb {B}^n)\) be a subclass of normalized biholomorphic mappings defined on the unit ball in \(\mathbb {C}^n,\) which is closely related to the starlike mappings. Firstly, we obtain the growth theorem for \(\widehat{\mathcal {S}}_g^{\alpha , \beta }(\mathbb {B}^n)\). Secondly, we apply the growth theorem and a new type of the boundary Schwarz lemma to establish the distortion theorems of the Fréchet-derivative type and the Jacobi-determinant type for this subclass, and the distortion theorems with g-starlike mapping (resp. starlike mapping) are partly established also. At last, we study the Kirwan and Pell type results for the compact set of mappings which have g-parametric representation associated with a modified Roper–Suffridge extension operator, which extend some earlier related results.

Keywords

Distortion estimates Extreme points Growth theorems Starlike mappings Support points 

Mathematics Subject Classification

32H02 30C45 

Notes

Acknowledgements

The project is supported by the National Natural Science Foundation of China (No. 11671306).

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsWuhan UniversityWuhanPeople’s Republic of China

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