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The Schwarz Type Lemmas and the Landau Type Theorem of Mappings Satisfying Poisson’s Equations

  • Shaolin ChenEmail author
  • David Kalaj
Article
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Abstract

For a given continuous function \(g:~\overline{\mathbb {D}}\rightarrow {\mathbb {C}}\) and a given continuous function \(\psi :~{\mathbb {T}}\rightarrow {\mathbb {C}}\), we establish some Schwarz type Lemmas for mappings f satisfying the PDE: \(\Delta f=g\) in \({\mathbb {D}}\), and \(f=\psi \) in \({\mathbb {T}}\), where \({\mathbb {D}}\) is the unit disk of the complex plane \({\mathbb {C}}\) and \({\mathbb {T}}=\partial {\mathbb {D}}\) is the unit circle. Then we apply these results to obtain a Landau type theorem, which is a partial answer to the open problem in Chen and Ponnusamy (Bull Aust Math Soc 97: 80–87, 2018).

Keywords

Schwarz’s Lemma Landau type theorem Poisson’s equation 

Mathematics Subject Classification

Primary 30H10 30C62 Secondary 31A05 31C05 

Notes

Acknowledgements

We thank the referee for providing constructive comments and help in improving this paper. This research was partly supported by the Science and Technology Plan Project of Hengyang City (No. 2018KJ125), the National Natural Science Foundation of China (No. 11571216), the Science and Technology Plan Project of Hunan Province (No. 2016TP1020), the Science and Technology Plan Project of Hengyang City (No. 2017KJ183), and the Application-Oriented Characterized Disciplines, Double First-Class University Project of Hunan Province (Xiangjiaotong [2018]469).

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.College of Mathematics and StatisticsHengyang Normal UniversityHengyangPeople’s Republic of China
  2. 2.Faculty of Natural Sciences and MathematicsUniversity of MontenegroPodgoricaMontenegro

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