An analogue of Koebe's theorem and the openness of a limit map in one class

TL;DR

This paper extends Koebe's theorem to a class of mappings satisfying a Poletsky-type inverse modulus inequality, demonstrating uniform image containment and the openness of limit maps under convergence.

Contribution

It introduces an analogue of Koebe's theorem for a specific class of mappings and proves the openness of limit maps in this class.

Findings

The image of a ball under these mappings contains a fixed ball uniformly.

Limit maps of converging sequences in this class are open.

The results generalize classical theorems for analytic functions.

Abstract

We study mappings that satisfy the inverse modulus inequality of Poletsky type in a fixed domain. It is shown that, under some additional restrictions, the image of a ball under such mappings contains a fixed ball uniformly over the class. This statement can be interpreted as the well-known analogue of Koebe's theorem for analytic functions. As an application of the obtained result, we show that, if a sequence of mappings belonging to the specified class converges locally uniformly, then the limit mapping is open.