On modulus inequality of the order $p$ for the inner dilatation

TL;DR

This paper investigates the relationship between Sobolev class mappings and modulus distortion bounds, proving inequalities related to inner dilatation of order p, with applications to homeomorphisms and mappings with branch points.

Contribution

It establishes a Poletsky-type inequality for inner dilatation of order p in mappings with bounded and finite distortion, extending understanding of modulus distortion bounds.

Findings

Proved Poletsky-type inequality for inner dilatation of order p.

Derived lower bounds for modulus distortion.

Analyzed cases of homeomorphisms and mappings with branch points.

Abstract

The article is devoted to mappings with bounded and finite distortion of plane domains. Our investigations are devoted to the connection between mappings of the Sobolev class and upper bounds for the distortion of the modulus of families of paths. For this class, we have proved the Poletsky-type inequality with respect to the so-called inner dilatation of the order~$p.$ Along the way, we also obtained lower bounds for the modulus distortion under mappings. We separately considered the situations of homeomorphisms and mappings with branch points.