Lappan's five-point theorem for ϕ-Normal Harmonic Mappings

Abstract

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

10 pages