The main purpose of this paper is twofold. First, a class of polyharmonic mappings with positive Jacobian and being univalent in the unit disk \({\mathbb {D}}\) are constructed. Second, we state and prove several theorems about the radius of univalence of a class of polyharmonic mappings, which gives a partial generalized answer to the radius problem posed by Abu Muhanna (Complex Var Elliptic Equ 53:745–751, 2008).

中文翻译：

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单价多谐映射的一个参数族

本文的主要目的是双重的。首先，构造一类具有正雅可比行列且在单位盘\（{\ mathbb {D}} \）中是单价的多谐映射。其次，我们陈述并证明关于一类多调和映射的非零半径的几个定理，这部分解决了由Abu Muhanna提出的半径问题的部分广义答案（Complex Var Elliptic Equ 53：745-751，2008）。