  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.
