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

The article is devoted to mappings with boundedand finite distortion of planar domains. Our investigations aredevoted to the connection between mappings of the Sobolev class andupper bounds for the distortion of the modulus of families of paths.For this class, we have proved the Poletsky-type inequality withrespect to the so-called inner dilatation of the order~$p.$ Weseparately considered the situations of homeomorphisms and mappingswith branch points. In particular, we have established thathomeomorphisms of the Sobolev class satisfy the upper estimate ofthe distortion of the modulus at the inner and boundary points ofthe domain. In addition, we have proved that similar estimates ofcapacity distortion occur at the inner points of the domain for opendiscrete mappings. Also, we have shown that open discrete and closedmappings satisfy some estimates of the distortion of the modulus offamilies of paths at the boundary points. The results of themanuscript are obtained mainly under the condition that theso-called inner dilatation of mappings is locally integrable. Themain approach used in the proofs is the choice of admissiblefunctions, using the relations between the modulus and capacity, andconnections between different modulus of families of paths (similarto Hesse, Ziemer and Shlyk equalities). In this context, we haveobtained some lower estimate of the modulus of families of paths inSobolev classes. The manuscript contains some examples related toapplications of obtained results to specific mappings.