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

### Abstract

The article is devoted to mappings with bounded

and finite distortion of planar 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~$p.$ We

separately considered the situations of homeomorphisms and mappings

with branch points. In particular, we have established that

homeomorphisms of the Sobolev class satisfy the upper estimate of

the distortion of the modulus at the inner and boundary points of

the domain. In addition, we have proved that similar estimates of

capacity distortion occur at the inner points of the domain for open

discrete mappings. Also, we have shown that open discrete and closed

mappings satisfy some estimates of the distortion of the modulus of

families of paths at the boundary points. The results of the

manuscript are obtained mainly under the condition that the

so-called inner dilatation of mappings is locally integrable. The

main approach used in the proofs is the choice of admissible

functions, using the relations between the modulus and capacity, and

connections between different modulus of families of paths (similar

to Hesse, Ziemer and Shlyk equalities). In this context, we have

obtained some lower estimate of the modulus of families of paths in

Sobolev classes. The manuscript contains some examples related to

applications of obtained results to specific mappings.

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*Matematychni Studii*,

*59*(2), 141-155. https://doi.org/10.30970/ms.59.2.141-155

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