Bohr-Type Inequalities for Shifted Disks via Optimal $H^2$-Embeddings

Abstract

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 $\Omega_\gamma$ parameterized by $\gamma \in [0, 1)$, defined by$$\Omega_\gamma = \left\{ z \in \mathbb{C} : \left| z + \frac{\gamma}{1 - \gamma} \right| < \frac{1}{1 - \gamma},\; \gamma\in [0, 1) \right\}.$$ By exploiting the geometric characteristics of $\Omega_\gamma$ and evaluating the limiting behavior as $\gamma \to 1^-$, 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.