Lappan’s five-point theorem for \(\varphi \)-normal harmonic mappings

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