On boundary distortion estimates of homeomorphisms with a fixed point

The article is devoted to the study of homeomorphisms that distort the modulus of families of paths according to the Poletsky inequality type. We consider the case when the majorant, corresponding to the distortion of the modulus, has finite mean oscillation at any point or satisfies Lehto integral divergence condition. We have obtained the boundary Hölder continuity of such mappings, provided that the image of at least one inner point is located at a fixed distance from the boundary of the mapped domain. In particular, this condition holds for fixed-point mappings. The manuscript deals with cases of good boundaries and domains with prime ends.