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Landau-Type Theorems for Certain Bounded Biharmonic Mappings

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Abstract

In this paper, we establish three sharp versions of the Landau-type theorems for bounded biharmonic mappings \(F(z)=|z|^2G(z)+H(z)\), where G(z) and H(z) are harmonic in the unit disk U with \(G(0)=H(0)=0\) and \(\lambda _{F}(0)=||F_z(0)|-|F_{{\overline{z}}}(0)||=1\). Our results generalize (or improve) the corresponding results given in Liu et al. (Math Methods Appl Sci 40:2582–2595, 2017). Three conjectures for the sharp version of the Landau-type theorem for certain bounded biharmonic mappings are given in third section.

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Acknowledgements

The research of the first author was supported by Guangdong Natural Science Foundation of China (No. 2018A030313508). The authors of this paper thank the referee very much for his valuable comments and suggestions to this paper.

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

    Ming-Sheng Liu & Li-Fang Luo

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  1. Ming-Sheng Liu
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  2. Li-Fang Luo
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Correspondence to Ming-Sheng Liu.

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Liu, MS., Luo, LF. Landau-Type Theorems for Certain Bounded Biharmonic Mappings. Results Math 74, 170 (2019). https://doi.org/10.1007/s00025-019-1095-7

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