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

Abstract

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

11 pages