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On Koebe's theorem for mappings with integral constraints

Evgeny Sevost'yanov, Valery Targonskii, Nataliya Ilkevych

Abstract

We study mappings that satisfy the inverse modulus inequality of Poletsky type with respect to $p$-modulus. Given $n-1<p\leqslant n,$ we show that, the image of some ball contains a fixed ball under mappings mentioned above. This statement can be interpreted as the well-known analogue of Koebe's theorem for analytic functions. As a consequence, we obtain the openness and discreteness of the limit mapping in the class under study. The paper also studies mappings of the Orlicz-Sobolev classes, for which an analogue of the Koebe one-quarter theorem is obtained as a consequence of the main results

On Koebe's theorem for mappings with integral constraints

Abstract

We study mappings that satisfy the inverse modulus inequality of Poletsky type with respect to -modulus. Given we show that, the image of some ball contains a fixed ball under mappings mentioned above. This statement can be interpreted as the well-known analogue of Koebe's theorem for analytic functions. As a consequence, we obtain the openness and discreteness of the limit mapping in the class under study. The paper also studies mappings of the Orlicz-Sobolev classes, for which an analogue of the Koebe one-quarter theorem is obtained as a consequence of the main results

Paper Structure

This paper contains 4 sections, 20 theorems, 54 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Lemmas
  4. Connection with Orlicz-Sobolev classes

Key Result

theorem 1.1

Let $n\geqslant 2,$$p\in (n-1, n]$ and let $D$ be a domain in ${\Bbb R}^n.$ Let also $\Phi:\overline{{\Bbb R^{+}}}\rightarrow \overline{{\Bbb R^{+}}}$ be an increasing convex function that satisfies the condition for some $\delta>\Phi(0).$ Assume that, the family $\frak{F}^{Q, p}_{a, b, \delta}(D)$ is equicontinuous at $a$ and $b.$ Then for every compactum $K$ in $D$ and for every $0<\varepsilon<

Figures (1)

  • Figure 1: To the proof of Lemma \ref{['lem1']}

Theorems & Definitions (29)

  • theorem 1.1
  • remark 1.1
  • remark 1.2
  • theorem 1.2
  • remark 1.3
  • proposition 2.1
  • lemma 2.1
  • proof
  • proposition 2.2
  • lemma 2.2
  • ...and 19 more