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Landau-type theorems and bi-Lipschitz theorems for bounded biharmonic mappings

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Abstract

In this paper, we first establish five versions of Landau-type theorems for five classes of bounded biharmonic mappings \(F(z)=|z|^2G(z)+H(z)\) on the unit disk \({\mathbb {D}}\) with \(G(0)=H(0)=J_F(0)-1=0\), which improve the related results of earlier authors. In particular, two versions of those Landau-type theorems are sharp. Then we derive five bi-Lipschitz theorems for these classes of bounded and normalized biharmonic mappings.

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The authors declare that this research is purely theoretical and does not associate with any datas.

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Acknowledgements

This research of the second author is partly supported by Guangdong Natural Science Foundations (Grant No. 2018A030313508).

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  1. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, Guangdong, People’s Republic of China

    Shi-Fei Chen & Ming-Sheng Liu

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  1. Shi-Fei Chen
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  2. Ming-Sheng Liu
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Correspondence to Ming-Sheng Liu.

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Communicated by Adrian Constantin.

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Chen, SF., Liu, MS. Landau-type theorems and bi-Lipschitz theorems for bounded biharmonic mappings. Monatsh Math 193, 783–806 (2020). https://doi.org/10.1007/s00605-020-01440-5

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