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On Montel theorem for mappings with inverse moduli inequalities

Miodrag Mateljevic, Evgeny Sevost'yanov

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.

On Montel theorem for mappings with inverse moduli inequalities

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.
Paper Structure (4 sections, 9 theorems, 85 equations, 5 figures)

This paper contains 4 sections, 9 theorems, 85 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The case where the family of mappings omits at least two points
  3. Proof of Theorem \ref{['th2']}
  4. Proof of Theorem \ref{['th1']}

Key Result

theorem 1.1

Assume that, the following condition holds: for each point $y_0\in {\Bbb R}^n$ there exists $r_0=r_0(y_0)>0$ such that, for every $0<r_1<r_2<r_0$ there is a set $E\subset[r_1, r_2]$ of a positive linear Lebesgue measure such that the function $Q$ is integrable with respect to $\mathcal{H}^{n-1}$ ove

Figures (5)

  • Figure 1: To the proof of Theorem \ref{['th3']}, Case 1.1
  • Figure 2: To the proof of Theorem \ref{['th3']}, Case 1.2
  • Figure 3: To the proof of Theorem \ref{['th3']}, Case 1.3
  • Figure 4: To the proof of Theorem \ref{['th3']}, Case 2
  • Figure 5: To the proof of Theorem \ref{['th1']}

Theorems & Definitions (14)

  • remark 1.1
  • theorem 1.1
  • corollary 1.1
  • theorem 1.2
  • corollary 1.2
  • proposition 2.1
  • lemma 2.1
  • lemma 2.2
  • theorem 2.3
  • proof
  • ...and 4 more