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Sharp Landau-Type Theorems and Schlicht Disc Radii for certain Subclasses of Harmonic Mappings

Molla Basir Ahamed, Rajesh Hossain

Abstract

Let $\mathcal{H}$ be the class of all complex-valued harmonic mappings $f=h+\overline{g}$ defined on the unit disc $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$ with the normalization $h(0)=0=h'(0)-1$, here $h$ and $g$ are analytic functions in $\mathbb{D}$. In this paper, we investigates Landau-type theorems for several significant subclasses of sense-preserving harmonic mappings. Specifically, we establish sharp Landau-type theorems for the class $\mathcal{P}_{\mathcal{H}}^{0}(M)$ and the parameterized class $\mathcal{W}_{\mathcal{H}}^{0}(α)$ for $α\ge 0$. For mappings in $\mathcal{W}_{\mathcal{H}}^{0}(α)$, we derive the radii of univalence and the radii of the largest schlicht discs contained in the images of the unit disc, expressing these results in terms of the Lerch Transcendent function $Φ(z,s,a)$ and the Dilogarithm function ${\rm Li}_2(z)$. The sharpness of the obtained radii is demonstrated by constructing appropriate extremal functions for each class. These results generalize and extend various known Landau-type theorems in the theory of harmonic mappings.

Sharp Landau-Type Theorems and Schlicht Disc Radii for certain Subclasses of Harmonic Mappings

Abstract

Let be the class of all complex-valued harmonic mappings defined on the unit disc with the normalization , here and are analytic functions in . In this paper, we investigates Landau-type theorems for several significant subclasses of sense-preserving harmonic mappings. Specifically, we establish sharp Landau-type theorems for the class and the parameterized class for . For mappings in , we derive the radii of univalence and the radii of the largest schlicht discs contained in the images of the unit disc, expressing these results in terms of the Lerch Transcendent function and the Dilogarithm function . The sharpness of the obtained radii is demonstrated by constructing appropriate extremal functions for each class. These results generalize and extend various known Landau-type theorems in the theory of harmonic mappings.

Paper Structure

This paper contains 6 sections, 6 theorems, 98 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Certain classes of harmonic mappings:
  3. Main results
  4. Landau theorem for harmonic mappings in the class $\mathcal{P}^0_{\mathcal{H}}(M)$
  5. Landau theorem for harmonic mappings in the class $\mathcal{W}_H^0(\alpha)$
  6. Landau theorem for functions in the class $\mathcal{G}_H^k(\alpha,1)$

Key Result

Theorem 2.1

Let $f\in\mathcal{P}_H^{0}(M)$ be a harmonic mapping on the unit disc $\mathbb{D}$ such that $f(0)=0, J_f(0)=1$ and $|f(z)|<M$ for $z\in\mathbb{D}$. Then, $f$ is univalent on a disc $\mathbb{D}_{\rho_1}$ with and $f(\mathbb{D}_{\rho_1})$ contains a schlicht disc with This result is sharp, with an extremal function given by

Figures (3)

  • Figure 1: Sharp geometric radii for mappings in $\mathcal{P}_{\mathcal{H}}^{0}(M)$. The left panels show the univalence disks $\mathbb{D}_{\rho_1}$ in the unit disk $\mathbb{D}$, and the right panels show the guaranteed schlicht disks $\mathbb{D}_{R_1}$. The plots illustrate the inverse relationship between the boundedness constant $M$ and the radii $\rho_1$ and $R_1$, emphasizing the sharp nature of the constants derived in Theorem \ref{['Th-2.1']}
  • Figure 2: Sharp Landau-type disks for the class $\mathcal{W}_{\mathcal{H}}^{0}(\alpha)$ corresponding to Corollaries 2.1 and 2.2. The left panels illustrate the domain plane, where the dashed circle represents the unit disk boundary $\mathbb{D}$ and the blue shaded regions denote the sharp univalence disks $\mathbb{D}_{\rho_2}$. The right panels illustrate the image plane, where the red shaded regions denote the guaranteed schlicht disks $\mathbb{D}_{R_2}$.
  • Figure 3: Geometric visualization of sharp Landau-type radii for the class $\mathcal{G}_{\mathcal{H}}^{k}(\alpha, 1)$ with $M=1.0$ across varying parameters $k \in \{1, 2, 3\}$ and $\alpha \in \{0.0, 0.5, 0.8\}$.

Theorems & Definitions (14)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.1
  • Remark 2.1
  • proof : Proof of Theorem \ref{['Th-2.1']}
  • Remark 2.2
  • Theorem 2.2
  • Corollary 2.1
  • Corollary 2.2
  • proof : Proof of Theorem \ref{['Th-2.2']}
  • ...and 4 more