A Unified Study of Bohr's Inequality for analytic and harmonic mappings on the Unit Disk

TL;DR

This work extends Bohr’s inequality beyond its classical analytic setting to a unified framework that covers analytic and harmonic mappings on the unit disk. It first strengthens the Bohr phenomenon for bounded analytic functions by incorporating the refined area term $S_r/(\pi-S_r)$, deriving a sharp radius with an explicit best-possible constant. It then generalizes Bohr-type bounds for harmonic mappings via a flexible sequence $\{\varphi_n(r)\}$, connecting radii to distance-from-boundary measures and establishing sharp results for classes like $\mathcal{P}^{0}_{\mathcal{H}}(M)$ and $\mathcal{W}^{0}_{\mathcal{H}}(\alpha)$, including special functions such as the Hurwitz–Lerch Zeta function. The findings unify several known Bohr-type results as special cases and provide new sharp radii and coefficient-growth controls with broad applicability in geometric function theory.

Abstract

We investigate improved forms of the Bohr inequality, using the quantity $S_r/π$, for analytic selfmaps in class $\mathcal{B}$ of $\mathbb{D}$, where $S_r$ is the area measure of $\mathbb{D}_r$. We then generalize the inequality for harmonic mappings ($\mathcal{P}^0_{\mathcal{H}}(M)$ and $\mathcal{W}^0_{\mathcal{H}}(α)$ of the form $f = h + \overline{g}$) by introducing a sequence $\{\varphi_n(r)\}_{n=0}^\infty$ of differentiable, increasing functions on $[0, 1)$. The Hurwitz Lerch Zeta function is utilized for some consequences, and all results are shown to be sharp.

Theorems & Definitions (9)

• Theorem 2.1
• Remark 2.1
• proof : Proof of Theorem \ref{['th-2.2']}
• Definition 3.1
• Theorem 3.1
• Remark 3.1
• Theorem 3.2
• Remark 3.2
• proof : Proof of Theorem \ref{['Th-4.1']}