On asymptotics of ring $Q$-homeomorphisms with respect to $p$-modulus near the origin

We consider the class of ring $Q$-homeomorphisms with respect to $p$-modulus in ${\mathbb{R}}^n$ with $p>n$, and obtain lower bounds for limsups of the distance distortions under such mappings. These estimates can be treated as Hölder’s continuity of the inverses near the origin. The sharpness is illustrated by example