On the inverse Poletsky inequality with cotangent dilatation

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