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

A harmonic mapping $f=h+\bar{g}$ in $\mathbb{D}$ is $\phi$-normal if $f^{\mathrm{\#}}(z)=\mathcal{O}(\mathrm{\mid }\phi (z)\mathrm{\mid }),\text{ as }\mathrm{\mid }z\mathrm{\mid }\to 1^{-},$ where $f^{\mathrm{\#}}(z)=(\mathrm{\mid }{h}^{\mathrm{\prime }}(z)\mathrm{\mid }+\mathrm{\mid }{g}^{\mathrm{\prime }}(z)\mathrm{\mid })\mathrm{/}(1+\mathrm{\mid }f(z){\mathrm{\mid }}^2)\mathrm{.}$ 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.