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Introducing a Novel Subclass of Harmonic Functions with Close-to-Convex Properties

Serkan Çakmak, Sibel Yalçin

TL;DR

This work introduces the harmonic subclass $\mathcal{KH}^{0}(k,\gamma)$ of $\mathcal{SH}^{0}$, defined by $\mathfrak{f}=\mathfrak{u}+\overline{\mathfrak{v}}$ with $0\leq \gamma<1$ and the condition $\mathrm{Re}\left\{\frac{z^k\mathfrak{u}'(z)}{\phi_k(z)}-\gamma\right\}>\left|\frac{z^k\mathfrak{v}'(z)}{\phi_k(z)}\right|$. It establishes a bridge to the analytic class $\mathcal{K}(k,\gamma)$ via the property that $\mathfrak{f}\in\mathcal{KH}^{0}(k,\gamma)$ iff $F_{\varepsilon}=\mathfrak{u}+\varepsilon\mathfrak{v}\in\mathcal{K}(k,\gamma)$ for all $|\varepsilon|=1$, and proves that $\mathcal{KH}^{0}(k,\gamma)$ is close-to-convex in the unit disk. The paper provides sharp coefficient bounds $|u_m|+|v_m|\le \gamma+m(1-\gamma)$ for $m\ge2$, derives distortion bounds, and shows that the class is closed under convex combinations and convolution. Concrete examples with $\phi(z)=z$ illustrate membership and resulting geometry of images (star-shaped vs convex), highlighting the framework's potential for extension to higher-order derivatives.

Abstract

In this paper, we introduce a new subclass of close-to-convex harmonic functions. We present a sufficient coefficient condition for a function to be a member of this class. Furthermore, we establish a distortion theorem. These results lay the groundwork for extending the findings to function classes involving higher-order derivatives.

Introducing a Novel Subclass of Harmonic Functions with Close-to-Convex Properties

TL;DR

This work introduces the harmonic subclass of , defined by with and the condition . It establishes a bridge to the analytic class via the property that iff for all , and proves that is close-to-convex in the unit disk. The paper provides sharp coefficient bounds for , derives distortion bounds, and shows that the class is closed under convex combinations and convolution. Concrete examples with illustrate membership and resulting geometry of images (star-shaped vs convex), highlighting the framework's potential for extension to higher-order derivatives.

Abstract

In this paper, we introduce a new subclass of close-to-convex harmonic functions. We present a sufficient coefficient condition for a function to be a member of this class. Furthermore, we establish a distortion theorem. These results lay the groundwork for extending the findings to function classes involving higher-order derivatives.

Paper Structure

This paper contains 4 sections, 9 theorems, 31 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Examples of Functions in the Class $\mathcal{KH}^{0}(k,\gamma)$
  3. Geometric Properties of the class $\mathcal{KH}^{0}(k,\gamma)$
  4. Conclusions

Key Result

Lemma 3.1

Let $\mathfrak{u}$ and $\mathfrak{v}$ be analytic functions in $\mathbb{E}$, such that $|\mathfrak{v}^{\prime}(0)| < |\mathfrak{u}^{\prime}(0)|$, and for each $\varepsilon$$(|\varepsilon| = 1)$, $F_{\varepsilon }=\mathfrak{u}+\varepsilon \mathfrak{v}$ is close-to-convex. Then, $\mathfrak{f}=\mathfra

Figures (6)

  • Figure 1: Under the map $\mathfrak{f} = z + \frac{99}{200}\overline{z}^2$, the image of the unit disk.
  • Figure 2: Under the map $\mathfrak{f} = z + \frac{1}{10}\overline{z}^2$, the image of the unit disk.
  • Figure 3: Under the map $\mathfrak{f} = z + \frac{33}{100}\overline{z}^3$, the image of the unit disk.
  • Figure 4: Under the map $\mathfrak{f} = z + \frac{1}{15}\overline{z}^3$, the image of the unit disk.
  • Figure 5: Under the map $\mathfrak{f} = z + \frac{99}{500}\overline{z}^5$, the image of the unit disk.
  • ...and 1 more figures

Theorems & Definitions (24)

  • Definition 1.1
  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • Example 2.7
  • Lemma 3.1
  • Theorem 3.2
  • ...and 14 more