New subclass of the class of close-to-convex harmonic mappings defined by a third-order differential inequality

TL;DR

This paper introduces a new subclass of harmonic functions defined by a third-order differential inequality, exploring its geometric properties, coefficient bounds, and stability under various operations.

Contribution

It defines a novel subclass of harmonic functions via a third-order differential inequality and analyzes its geometric and algebraic properties, including convexity, starlikeness, and closure under operations.

Findings

The class is close-to-convex.

Coefficient bounds are established.

The class is closed under convex combination and convolution.

Abstract

In this paper, we introduce a new subclass of harmonic functions $f=s+\overline{t}$ in the open unit disk $U =\left \{ z\in C:\left \vert z\right \vert <1\right \} $ satisfying ${\text{Re}}\left[ \gamma s^{\prime }(z)+\delta zs^{\prime \prime }(z)+\left( \frac{\delta -\gamma }{2}\right) z^{2}s^{\prime \prime \prime }\left( z\right) -\lambda \right] >\left \vert \gamma t^{\prime }(z)+\delta zt^{\prime \prime }(z)+\left( \frac{\delta -\gamma }{2}\right) z^{2}t ^{\prime \prime \prime }\left( z\right) \right \vert,$ where $0\leq \lambda <\gamma \leq \delta, z\in U.$ We determine several properties of this class such as close-to-convexity, coefficient bounds, and growth estimates. We also prove that this class is closed under convex combination and convolution of its members. Furthermore, we investigate the properties of fully starlikeness and fully convexity of the class.