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On Landau-Type Theorems for Poly-Analytic Functions

Vasudevarao Allu, Rohit Kumar

Abstract

In this paper, we establish three Landau-type theorems for certain bounded poly-analytic functions, which generalize the corresponding result for bi-analytic functions given by Liu and Ponnusamy [Canad. Math. Bull. 67(1): 2024, 152-165]. Further, we prove three bi-Lipschitz theorems for these subclasses of poly-analytic functions.

On Landau-Type Theorems for Poly-Analytic Functions

Abstract

In this paper, we establish three Landau-type theorems for certain bounded poly-analytic functions, which generalize the corresponding result for bi-analytic functions given by Liu and Ponnusamy [Canad. Math. Bull. 67(1): 2024, 152-165]. Further, we prove three bi-Lipschitz theorems for these subclasses of poly-analytic functions.
Paper Structure (5 sections, 15 theorems, 67 equations)

This paper contains 5 sections, 15 theorems, 67 equations.

Table of Contents

  1. Introduction and Preliminaries
  2. The Landau Theorem
  3. Key Results
  4. Landau-Bloch type Theorem for Poly-analytic functions
  5. The Bi-Lipschitz Theroems for Some Poly-analytic functions

Key Result

Theorem A

Chen-Gauthier-Hengertner-2000 Let $f$ be a harmonic mapping of the unit disc $\mathbb{D}$ such that $f(0)=0$, $f_{\bar{z}}(0)=0, f_z(0)=1$, and $|f(z)|<M$ for $z \in \mathbb{D}$. Then, $f$ is univalent on a disc $\mathbb{D}_{\rho_0}$ with and $f\left(\mathbb{D}_{\rho_0}\right)$ contains a schlicht disc $\mathbb{D}_{R_0}$ with where $m \approx 6.85$ is the minimum of the function $(3-r^2)/(r(1-r^

Theorems & Definitions (22)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem E
  • Lemma 2.1
  • Lemma 2.2
  • Theorem F
  • Lemma 2.3
  • proof
  • ...and 12 more