A sufficient condition for $p$-valently harmonic functions

TL;DR

This paper establishes a sufficient condition on the analytic part of harmonic functions to ensure they are p-valent in the unit disk, supported by examples and visualizations.

Contribution

It introduces a new sufficient condition for p-valency of harmonic functions based on their analytic component, expanding understanding of harmonic function behavior.

Findings

Derived a sufficient condition for p-valency of harmonic functions

Provided examples illustrating the condition and resulting images

Enhanced understanding of harmonic functions in the unit disk

Abstract

For normalized harmonic functions $f(z)=h(z)+\bar{g(z)}$ in the open unit disk $\mathbb{U}$, a sufficient condition on $h(z)$ for $f(z)$ to be $p$-valent in $\mathbb{U}$ is discussed. Moreover, some interesting examples and images of $f(z)$ satisfying the obtained condition are enumerated.