On the inverse Poletsky inequality with cotangent dilatation

TL;DR

This paper establishes an inverse Poletsky inequality involving cotangent dilatation for wide classes of mappings with branch points, revealing new relationships between dilatations and mapping distortions.

Contribution

It introduces the inverse Poletsky inequality with cotangent dilatation for mappings with branch points and relates it to tangential dilatation, expanding understanding of mapping distortions.

Findings

Cotangent dilatation coincides with tangential dilatation for inverse mappings.

Cotangent dilatation can be less than one on sets of positive measure.

The inequality provides bounds on the distortion of path families in complex mappings.

Abstract

The article is devoted to establishing the distortion of the modulus of families of paths in wide classes of mappings that admit branch points. In particular, for mappings that are differentiable almost everywhere and have $N$- and $N^{\,- 1}$-Luzin properties and are absolutely continuous on almost all paths, we obtained the inverse Poletsky inequality with the so-called cotangent dilatation. We have proved that, for inverse mappings, this dilatation coincides with the so-called tangential dilatation of the corresponding inverse mapping. In addition, we have proved that cotangent dilatation is less than the outher or inner dilatation, in particular, may be less than one on the set of positive Lebesgue measure