Geometric analysis of a class of harmonic mappings defined by a differential inequality

TL;DR

This paper introduces and systematically investigates a new class of normalized harmonic mappings defined by a differential inequality, extending existing theories and providing sharp bounds and geometric properties.

Contribution

It defines a unified class of harmonic mappings based on a differential inequality, deriving sharp bounds and geometric properties, and exploring closure properties.

Findings

Established sharp coefficient bounds for the class.

Determined radii of univalency, starlikeness, and convexity.

Proved closure under convex combinations and convolution under certain conditions.

Abstract

In this paper, we introduces and undertake as a systematical investigation of the class $\mathcal{P}_{\mathcal{H}}^{0}(\alpha,M)$ of normalized harmonic mappings $f = h + \overline{g}$ in the unit disk $\mathbb{D}$, defined by the differential inequality \[ \text{Re}\left((1-\alpha)h'(z) + \alpha z h''(z)\right) > -M + \left|(1-\alpha)g'(z) + \alpha z g''(z)\right|\quad\text{for}\quad z\in\Bbb{D}, \] where $M > 0$, $\alpha \in (0,1]$, and $g'(0) = 0$. This class extends the harmonic analogue of functions with positive real part and offers a unified framework for analyzing their geometric characteristics. We obtain sharp coefficient bounds for both the analytic and co-analytic parts, establish sharp growth bounds, and determine the radii of univalency, starlikeness, and convexity. Furthermore, we show that $\mathcal{P}_{\mathcal{H}}^{0}(\alpha,M)$ is closed under convex combinations, and under suitable restrictions on the parameters, it is also closed under convolution. Our findings generalize and extend several known results in the theory of harmonic mappings.