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We  prove  a quasiconformal analogue of Koebe's theorem  related to the average Jacobian and  use a normal family argument here to  prove a quasiregular analogue of this result in certain domains in $n$-dimensional space.
As an application, we  establish that Lipschitz-type properties are inherited by a quasiregular  function from its modulu.  We also prove some results of Hardy- Littlewood type  for Lipschitz-type spaces in several dimensions,  give  the characterization of Lipschitz-type spaces for quasiquasiregular mappings  by the average Jacobian  and give a short review of the subject. In particular, we solve so called  Dyakonov's problem.

Quasiregular, the harmonic analogue of Koebe’s one-quarter theorem, Lipschitz-type spaces, Hardy-Littlewood property