A Zalcman-Pang type rescaling result for $\phi$-normal harmonic mappings and applications

We investigate normality and $\phi$-normality criteria for harmonic mappings in the unit disk. A new sufficient condition for normality involving extended spherical derivatives is established. We further prove a Zalcman-Pang type rescaling lemma for $\phi$-normal harmonic mappings and derive new $\phi$-normality criteria as applications. In addition, we introduce $\phi$-normal families of harmonic mappings and obtain corresponding Lappan-type characterizations. In particular, we show that for sense-preserving harmonic mappings the relevant test set may be taken to consist of only three points.