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.

中文翻译：

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用于有界双调和映射的朗道型定理和双李普希茨定理

在本文中，我们首先在单元盘 $${ 上为五类有界双调和映射 $$F(z)=|z|^2G(z)+H(z)$$ 建立五个版本的朗道型定理\mathbb {D}}$$ 与 $$G(0)=H(0)=J_F(0)-1=0$$ ，改进了早期作者的相关结果。特别是，这些朗道式定理的两个版本是尖锐的。然后我们为这些有界和归一化的双调和映射类推导出五个双李普希茨定理。