涉及面积测度的近凸调和映射的尖锐Bohr与Bohr-Rogosinski不等式
Sharp Bohr and Bohr-Rogosinski inequalities involving area measure for close-to-convex harmonic mappings
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中文总结 AI 辅助
本文针对近凸调和映射子类，通过引入与面积积分相关的单调函数，建立了用欧几里得距离表示的尖锐Bohr型不等式及含模幂的Bohr-Rogosinski型不等式，并证明半径尖锐性及识别极值函数，推广统一了多个经典结果。
中文摘要 AI 辅助
本文研究了定义在开单位圆盘$\mathbb{D} \subset \mathbb{C}$上的单叶近凸调和映射$f = h + \overline{g}$的归一化子类$\mathcal{P}_{\mathcal{H}}^{0}(\alpha)$（$0 \le \alpha < 1$）的Bohr与Bohr-Rogosinski不等式的精细化与推广版本。通过引入与像域$f(\mathbb{D}_r)$的平面面积积分$S_r/\pi$相关的非负单调递增函数，我们建立了用欧几里得距离$d(f(0), \partial f(\mathbb{D}))$表示的新的尖锐Bohr型不等式。此外，我们建立了涉及映射模的幂$|f(z)|^p$（$p \ge 1$）的尖锐Bohr-Rogosinski型不等式。所有相关半径均被证明是尖锐的，并且明确识别了实现等号情形的极值函数。作为应用，我们的结果推广并统一了几何函数论中若干著名的经典定理和近期定理。
英文摘要
In this article, we investigate refined and generalized versions of the Bohr and Bohr--Rogosinski inequalities for a normalized subclass $\mathcal{P}_{\mathcal{H}}^{0}(α)$ ($0 \le α< 1$) of univalent close-to-convex harmonic mappings $f = h + \overline{g}$ defined on the open unit disk $\mathbb{D} \subset \mathbb{C}$. By incorporating non-negative monotone increasing functions associated with the planar area integral $S_r/π$ of the image domain $f(\mathbb{D}_r)$, we establish new sharp Bohr-type inequalities expressed in terms of the Euclidean distance $d(f(0), \partial f(\mathbb{D}))$. Furthermore, we establish sharp Bohr--Rogosinski-type inequalities involving powers of the modulus of the mapping $|f(z)|^p$ ($p \ge 1$). All associated radii are proven to be sharp, and extremal functions realizing the equality cases are explicitly identified. As applications, our results generalize and unify several well-known classical and recent theorems in geometric function theory.
发表机构
- Jadavpur University（贾达普大学）
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