On $\phi$-Normality of Harmonic Mappings and Their Families

Abstract

In this paper, we present Lappan's five-point type criteria for $\phi$-normality of sense-preserving harmonic mappings and families of harmonic mappings in the unit disc $\mathbb{D}$. We establish a normality criterion involving the higher-order spherical derivative of order $k$, and $k+2$ distinct test values under suitable conditions. We further reduce the number of test values to $\left[\frac{k}{2}\right]+2$ under an additional differential condition involving the analytic and co-analytic parts. We also prove a generalization in which the spherical derivative is replaced by a polynomial with non-negative real coefficients.