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

Abstract

We investigate normality and $ϕ$-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 $ϕ$-normal harmonic mappings and derive new $ϕ$-normality criteria as applications. In addition, we introduce $ϕ$-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.