The sharp refined Bohr inequalities for a subclass of close-to-convex harmonic mappings

TL;DR

This paper establishes sharp refined Bohr inequalities, improved Bohr radius, and Bohr-Rogosinski inequalities for a specific subclass of close-to-convex harmonic mappings defined by certain differential inequalities.

Contribution

It introduces new sharp bounds and inequalities for a subclass of harmonic functions characterized by differential inequalities, extending classical Bohr phenomena.

Findings

Derived sharp improved Bohr inequalities for the class.

Established refined Bohr radius for the subclass.

Proved Bohr-Rogosinski inequalities specific to the class.

Abstract

Let $\mathcal{H}$ be the class of normalized complex valued harmonic functions $ f = h + \overline{g}$ defined on the unit disk $\mathbb{D}$, where $h$ and $g$ are analytic functions with the normalization conditions $h(0) = h'(0) - 1 = 0$ and $g(0) = 0$. For the class $R_H^{0}(\gamma, \delta, \lambda)$ ( $0 \leq \lambda < \gamma \leq \delta$) consisting of functions \( f = h+\bar{g} \in \mathcal{H}\) satisfying the condition $f_{\overline{z}}(0)=0$ and the inequality $ Re(\gamma h'(z)+\delta z h''(z) +(\frac{\delta - \gamma}{2})z^2 h'''(z)-\lambda)> |\gamma g'(z)+\delta z g''(z) +(\frac{\delta - \gamma}{2})z^2 g'''(z)|$, we obtain sharp improved Bohr Phenomenon, refined Bohr radius and the Bohr-Rogosinski inequality for the class $R_H^{0}(\gamma, \delta, \lambda)$.