AI 中文总结
本文通过引入参数化平移圆盘族，结合其几何特征与极限分析建立新框架，确定无界半平面的玻尔半径并证明多类精确玻尔不等式，拓展了经典半径问题的适用范围并揭示区域形变与系数估计的关联。
AI 中文摘要
本文的主要目标是系统推广这一现象，将标准单位圆盘替换为一族嵌套、内切的平移圆盘$Ω_γ$，其参数为$γ\in [0, 1)$，定义为$$Ω_γ= \left\{ z \in \mathbb{C} : \left| z + \fracγ{1 - γ} \right| < \frac{1}{1 - γ},\\; γ\in [0, 1) \right\}.$$通过利用$Ω_γ$的几何特征并分析$γ\to 1^-$时的极限行为，我们建立了一个新的框架来确定无界半平面$\mathbb{H}_1 = \{z \in \mathbb{C} : \text{Re}(z) < 1\}$的玻尔半径。此外，我们证明了这些区域内玻尔不等式的若干精确变体，包括针对幺模有界解析函数的精细化改进形式。本文所得结果不仅将经典半径问题推广到无界区域，还揭示了区域形变与系数估计之间的微妙相互作用。
英文摘要
The primary objective of this paper is to systematically generalize this phenomenon by replacing the standard unit disk with a family of nested, internally tangent shifted disks $Ω_γ$ parameterized by $γ\in [0, 1)$, defined by$$Ω_γ= \left\{ z \in \mathbb{C} : \left| z + \fracγ{1 - γ} \right| < \frac{1}{1 - γ},\; γ\in [0, 1) \right\}.$$ By exploiting the geometric characteristics of $Ω_γ$ and evaluating the limiting behavior as $γ\to 1^-$, we establish a novel framework to determine the Bohr radius for the unbounded half-plane $\mathbb{H}_1 = \{z \in \mathbb{C} : \text{Re}(z) < 1\}$. Furthermore, we prove several sharp variations of the Bohr inequality within these domains, including refined and improved formulations for unimodular bounded analytic functions. The results obtained herein not only extend classical radius problems to unbounded regions but also illuminate the delicate interplay between domain deformation and coefficient estimates.
Comments23 pages, 4 figures
