Lappan's five-point theorem for {\phi}-Normal Harmonic Mappings

TL;DR

This paper investigates conditions under which harmonic mappings are -normal and extends Lappan's five-point theorem to -normal harmonic mappings, contributing to the understanding of their boundary behavior.

Contribution

It introduces new sufficient conditions for -normality and generalizes Lappan's five-point theorem to -normal harmonic mappings.

Findings

Established several sufficient conditions for -normal harmonic mappings.

Extended Lappan's five-point theorem to -normal harmonic mappings.

Provided insights into boundary behavior of harmonic mappings.

Abstract

A harmonic mapping $f=h+\overline{g}$ in $\mathbb{D}$ is $\varphi$-normal if $f^{\#}(z)=\mathcal{O}(|\varphi(z)|), \text{ as } |z|\to 1^-,$ where $f^{\#}(z)={(|h'(z)|+|g'(z)|)}/{(1+|f(z)|^2)}.$ In this paper, we establish several sufficient conditions for harmonic mappings to be $\varphi$-normal. We also extend the five-point theorem of Lappan for $\varphi$-normal harmonic mappings.