In this paper, we investigate the Bohr radius for $K$-quasiregular sense-preserving harmonic mappings $f=h+\overline{g}$ in the unit disk $\mathbb{D}$ such that the translated analytic part $h(z)-h(0)$ is quasi-subordinate to some analytic function. The main aim of this article is to extend and to establish sharp versions of four recent theorems by Liu and Ponnusamy \cite{LP2019} and, in particular, we settle affirmatively the two conjectures proposed by them. Furthermore, we establish two refined versions of Bohr's inequalities and determine the Bohr radius for the derivatives of analytic functions associated with quasi-subordination.

中文翻译：

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拟从属和 K-拟正则调和映射类的玻尔现象

在本文中，我们研究了单位圆盘 $\mathbb{D}$ 中 $K$-拟正则守义调和映射 $f=h+\overline{g}$ 的玻尔半径，使得翻译后的解析部分 $h( z)-h(0)$ 准从属于某个解析函数。本文的主要目的是扩展和建立 Liu 和 Ponnusamy \cite{LP2019} 最近的四个定理的尖锐版本，特别是我们肯定地解决了他们提出的两个猜想。此外，我们建立了玻尔不等式的两个改进版本，并确定了与准从属相关的解析函数的导数的玻尔半径。