Results on normal harmonic and $\varphi$-normal harmonic mappings

TL;DR

This paper investigates the properties of normal and $$-normal harmonic mappings, establishing new criteria and extending classical results like the Zalcman-Pang lemma to harmonic functions.

Contribution

It introduces the harmonic mapping analogue of the Zalcman-Pang lemma and defines the extended spherical derivative for harmonic mappings, providing new conditions for $$-normality.

Findings

Harmonic mappings with bounded spherical derivative are normal.

Extended spherical derivative concept aids in characterizing $$-normal harmonic mappings.

New sufficient conditions for $$-normality in harmonic mappings.

Abstract

In this paper, we study the concepts of normal functions and $\varphi$-normal functions in the framework of planar harmonic mappings. We establish the harmonic mapping counterpart of the well-known Zalcman-Pang lemma and as a consequence, we prove that a harmonic mapping whose spherical derivative is bounded away from zero is normal. Furthermore, we introduce the concept of the extended spherical derivative for harmonic mappings and obtain several sufficient conditions for a harmonic mapping to be $\varphi$-normal.