Normality Criteria for Quasimeromorphic Mappings: Avoidance and Sharing

Abstract

We establish normality criteria for families of quasimeromorphic mappings in \(\mathbb{R}^n\), \(n\geq 2\), based on the avoidance and sharing of continuous functions. We first establish a rescaling principle for sequences of continuous functions and use it to derive a higher-dimensional analogue of Lappan's avoidance criterion. We then obtain normality criteria based on the sharing of continuous functions. In particular, we prove that a family of \(K\)-quasimeromorphic mappings sharing \(q_0(n,K)+1\) mutually disjoint continuous functions is normal, where \(q_0(n,K)\) is the corresponding quasiregular Picard constant. We further consider sequences of quasimeromorphic mappings sharing sequences of continuous functions with a normal comparison sequence and establish a corresponding normality criterion, together with an improved bound under an additional separation condition on the limiting functions. Finally, we derive applications to normal quasiregular mappings into \(\mathbb{S}^n\).