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On the univalence criteria for elliptic polyharmonic and polyelliptic-harmonic mappings

Rajib Mandal, Sudip Kumar Guin

Abstract

In this paper, we first establish Landau-Bloch-type theorems for poly $(K,K')$-elliptic harmonic mappings, which are sharp in some given cases. Thereafter, we provide several coefficient bounds for $(K,K')$-elliptic and $K$-quasiregular polyharmonic mappings with bounded minimum distortion. Furthermore, using these coefficient bounds, we establish Landau-Bloch-type theorems for these mappings.

On the univalence criteria for elliptic polyharmonic and polyelliptic-harmonic mappings

Abstract

In this paper, we first establish Landau-Bloch-type theorems for poly -elliptic harmonic mappings, which are sharp in some given cases. Thereafter, we provide several coefficient bounds for -elliptic and -quasiregular polyharmonic mappings with bounded minimum distortion. Furthermore, using these coefficient bounds, we establish Landau-Bloch-type theorems for these mappings.
Paper Structure (3 sections, 12 theorems, 91 equations)

This paper contains 3 sections, 12 theorems, 91 equations.

Table of Contents

  1. Introduction and Preliminaries
  2. Main Results
  3. Proofs of the Main Results

Key Result

Theorem 2.1

Let $F(z)=\sum_{k=1}^p |z|^{2(k-1)}G_{p-k+1}(z)$ be a poly $(K,K')$-elliptic harmonic mapping in $\mathbb{D}$ such that $F(0)=0$, and satisfying the following conditions: Then $F(z)$ is univalent in $\mathbb{D}_{r_1}$, where $r_1$ is the unique root in $(0,1)$ of the equation where $\phi(r)$ is given by (1z1) and $\Lambda_p'=\left(K\Lambda_p+\sqrt{K^2\Lambda_p^2+4K'}\right)/2$. Furthermore, $F(\

Theorems & Definitions (27)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Corollary 2.1
  • Theorem 2.4
  • Corollary 2.2
  • Theorem 2.5
  • ...and 17 more