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The Landau-Bloch type theorems for certain class of holomorphic and pluriharmonic mappings in $\mathbb{c}^n$

Vasudeva Rao Allu, Rohit Kumar

Abstract

In this paper, we first define two classes of holomorphic mappings defined on the unit ball $B^n$ of n-dimensional complex space $\mathbb{C}^n$ and obtain the lower estimates for Bloch's constant for these classes. Also, we derive the Landau-Bloch type theorem for some subclasses of pluriharmonic mappings defined on the unit ball $B^n$.

The Landau-Bloch type theorems for certain class of holomorphic and pluriharmonic mappings in $\mathbb{c}^n$

Abstract

In this paper, we first define two classes of holomorphic mappings defined on the unit ball of n-dimensional complex space and obtain the lower estimates for Bloch's constant for these classes. Also, we derive the Landau-Bloch type theorem for some subclasses of pluriharmonic mappings defined on the unit ball .
Paper Structure (4 sections, 12 theorems, 103 equations)

This paper contains 4 sections, 12 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. The Bloch theorem for Bochner $(K,K')$-mappings
  3. The Bloch theorem for $(K,K')$-quasiregular mappings
  4. The Landau-Bloch type theorem for pluriharmonic mappings

Key Result

Theorem A

Bloch-1925 Let $f$ be a holomorphic function on the $\overline{\mathbb{D}}=\{z:|z|\leq 1\}$ and satisfying $|f'(0)|=1$. Then there exists a positive constant $b$ such that $f(\mathbb{D})$ contains a schlicht disc of radius $b$.

Theorems & Definitions (15)

  • Theorem A
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem B
  • Lemma A
  • Theorem 1
  • Corollary 1
  • Lemma B
  • Theorem 2
  • ...and 5 more