Landau-Bloch type theorem for elliptic and $K$-quasiregular harmonic mappings

TL;DR

This paper improves coefficient bounds for quasiregular and elliptic harmonic mappings and establishes Landau-Bloch type theorems for these classes, including $K$-quasiconformal harmonic self maps.

Contribution

It introduces new coefficient bounds and Landau-Bloch type theorems for elliptic and $K$-quasiregular harmonic mappings in the plane.

Findings

Established improved coefficient bounds for quasiregular and elliptic harmonic mappings.

Proved Landau-Bloch type theorems for $(K,K')$-elliptic and $K$-quasiregular harmonic mappings.

Derived coefficient estimates for $K$-quasiconformal harmonic self maps on the unit disk.

Abstract

In this paper, we establish an improved coefficient bounds for quasiregular and elliptic harmonic mappings and using these bounds we establish Landau-Bloch type theorem for $(K,K')$-elliptic and K-quasiregular harmonic mappings in plane. Furthermore, we prove the coefficient estimates for $K$-quasiconformal harmonic self maps defined on the unit disk $\mathbb{D}$.