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).
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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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Communicated by Vladimir Bolotnikov.
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Chen, S., Kalaj, D. The Schwarz Type Lemmas and the Landau Type Theorem of Mappings Satisfying Poisson’s Equations. Complex Anal. Oper. Theory 13, 2049–2068 (2019). https://doi.org/10.1007/s11785-019-00911-4
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DOI: https://doi.org/10.1007/s11785-019-00911-4