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On the inclusion properties for harmonic error functions

Şahsene Altınkaya, Sibel Yalçın

TL;DR

The paper studies inclusion properties for harmonic error functions derived from the analytic error function via convolution. It defines the harmonic error-function class $\mathcal{E}S_{\mathcal{H}}(k,\lambda,\gamma)$ and its co-analytic counterpart, and derives a coefficient-based condition that guarantees sense-preserving, harmonic univalence in the unit disk. It provides necessary and sufficient criteria for membership in the conjugate class, along with distortion bounds, extreme points, and convex-closure properties, and proves invariance under the generalized Bernardi-Libera-Livingston operator. These results advance geometric function theory for harmonic mappings associated with error-function convolutions and offer concrete tools for verifying univalence and related geometric properties.

Abstract

For the error functions of the form \begin{equation*} E_{r}\mathfrak{f}(z)=\frac{\sqrt{πz}}{2}er\ \mathfrak{f}(\sqrt{z})=z+Σ_{n=2}^{\infty} \frac{(-1)^{n-1}}{(2n-1)(n-1)!}z^{n}, \end{equation*}% let $\mathcal{E}S_{\mathcal{H}}(k,λ,γ)\,$\ 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.

On the inclusion properties for harmonic error functions

TL;DR

The paper studies inclusion properties for harmonic error functions derived from the analytic error function via convolution. It defines the harmonic error-function class and its co-analytic counterpart, and derives a coefficient-based condition that guarantees sense-preserving, harmonic univalence in the unit disk. It provides necessary and sufficient criteria for membership in the conjugate class, along with distortion bounds, extreme points, and convex-closure properties, and proves invariance under the generalized Bernardi-Libera-Livingston operator. These results advance geometric function theory for harmonic mappings associated with error-function convolutions and offer concrete tools for verifying univalence and related geometric properties.

Abstract

For the error functions of the form \begin{equation*} E_{r}\mathfrak{f}(z)=\frac{\sqrt{πz}}{2}er\ \mathfrak{f}(\sqrt{z})=z+Σ_{n=2}^{\infty} \frac{(-1)^{n-1}}{(2n-1)(n-1)!}z^{n}, \end{equation*}% let \ represent the class of harmonic error functions in the open unit disk . The paper attempts to present some basic properties for functions in this class.
Paper Structure (2 sections, 7 theorems, 43 equations)

This paper contains 2 sections, 7 theorems, 43 equations.

Table of Contents

  1. Introduction and background
  2. Main results

Key Result

Theorem 1

If a function $f\in \mathcal{H}$ of the form $(C)$ fulfills then $f$ is sense-preserving, harmonic univalent in $\mathbb{U}$ and $f\in \mathcal{E}S_{\mathcal{H}}(k,\lambda ,\gamma )$.

Theorems & Definitions (13)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Corollary 1
  • Theorem 4
  • proof
  • Theorem 5
  • ...and 3 more