Equicontinuity and normality of mappings with integrally bounded $p$-moduli

Abstract

We consider the generic discrete open mappings in ${\mathbb{R}}^n$ under which the perturbation of extremal lengths of curve collections is controlled integrally via $\int Q(x){\eta }^p(|x-x_0|)dm(x)$ with $n-1<p<n$, where $Q$ is a measurable function on ${\mathbb{R}}^n$ and $\underset{r_1}{\overset{r_2}{\int }}\eta (r)dr\ge 1$ for any $\eta$ on a given interval $[r_1,r_2].$ We proved that the family of all open discrete mappings of above type is normal under appropriate restrictions on the majorant $Q.$