Bohr radius for certain close-to-convex harmonic mappings

TL;DR

This paper investigates the Bohr-Rogosinski and related radii for a specific class of close-to-convex harmonic functions in the unit disk, providing sharp and improved bounds.

Contribution

It derives the sharp Bohr-Rogosinski, improved Bohr, and refined Bohr radii for the class of close-to-convex harmonic mappings defined by Li and Ponnusamy.

Findings

Sharp Bohr-Rogosinski radius obtained

Improved Bohr radius established

Refined Bohr radius determined

Abstract

Let $ \mathcal{H} $ be the class of harmonic functions $ f=h+\bar{g} $ in the unit disk $\mathbb{D}:=\{z\in\mathbb{C} : |z|<1\}$, where $ h $ and $ g $ are analytic in $ \mathbb{D} $. Let $$\mathcal{P}_{\mathcal{H}}^{0}(\alpha)=\{f=h+\overline{g} \in \mathcal{H} : \real (h^{\prime}(z)-\alpha)>|g^{\prime}(z)|\; \mbox{with}\; 0\leq\alpha<1,\; g^{\prime}(0)=0,\; z \in \mathbb{D}\} $$ be the class of close-to-convex mappings defined by Li and Ponnusamy \cite{Injectivity section}. In this paper, we obtain the sharp Bohr-Rogosinski radius, improved Bohr radius and refined Bohr radius for the class $ \mathcal{P}_{\mathcal{H}}^{0}(\alpha) $.