On the inclusion properties for harmonic error functions

TL;DR

This paper investigates the inclusion properties of harmonic error functions within a specific class in the unit disk, analyzing their fundamental characteristics and establishing foundational properties.

Contribution

It introduces and explores basic properties of harmonic error functions in a defined class, expanding understanding of their behavior in complex analysis.

Findings

Characterization of harmonic error functions in the class

Basic inclusion properties established

Foundational properties for further research

Abstract

For the error functions of the form \begin{equation*} E_{r}\mathfrak{f}(z)=\frac{\sqrt{\pi z}}{2}er\ \mathfrak{f}(\sqrt{z})=z+\Sigma_{n=2}^{\infty} \frac{(-1)^{n-1}}{(2n-1)(n-1)!}z^{n}, \end{equation*}% let $\mathcal{E}S_{\mathcal{H}}(k,\lambda ,\gamma )\,$\ represent the class of harmonic error functions $\mathcal{ERF}=\mathcal{ERH}+\overline{\mathcal{% ERG}}$ in the open unit disk $\mathbb{U}=\left\{ z\in \mathbb{C}:\ \ \left\vert z\right\vert <1\right\} $. The paper attempts to present some basic properties for functions in this class.