On Koebe's theorem for mappings with integral constraints

Abstract

We study mappings that satisfy the inverse modulus inequality of Poletsky type with respect to $p$-modulus. Given $n-1<p\leqslant n,$ we show that, the image of some ball contains a fixed ball under mappings mentioned above. This statement can be interpreted as the well-known analogue of Koebe's theorem for analytic functions. As a consequence, we obtain the openness and discreteness of the limit mapping in the class under study. The paper also studies mappings of the Orlicz-Sobolev classes, for which an analogue of the Koebe one-quarter theorem is obtained as a consequence of the main results