On Montel theorem for mappings with inverse moduli inequalities

TL;DR

This paper generalizes Montel's theorem by establishing conditions for the equicontinuity of families of mappings with inverse moduli inequalities, specifically those satisfying the inverse Poletskii inequality.

Contribution

It introduces new criteria for equicontinuity of mappings with finite distortion, extending Montel's theorem to a broader class of functions.

Findings

Families of mappings omitting at least one point are equicontinuous under integrability conditions.

The results generalize Montel's theorem to mappings satisfying inverse Poletskii inequality.

Analytic functions with finite multiplicity satisfy the inverse Poletskii inequality.

Abstract

This paper is devoted to the study of mappings with finite distortion, in particular, mappings satisfying the inverse Poletskii inequality. We study the problem of equicontinuity of families of such mappings in a given domain. We establish that a family of open discrete mappings with the inverse Poletskii inequality, omitting at least one point, is equicontinuous if the majorant responsible for the distortion of the modulus of families of paths under the mapping is integrable over almost all concentric spheres centered at the given point. Since analytic functions with finite multiplicity satisfy the inverse Poletskii inequality, this result generalizes the well-known Montel theorem on the normality of families.