  Suppose that $f: \bR^n\to\bR^n$ is a mapping of $K$-bounded $p$-mean
distortion for some $p>n-1$. We prove the equivalence of the following
properties of $f$: doubling condition for $J(x,f)$ over big balls centered at
origin, boundedness of multiplicity function $N(f,\bR^n)$, polynomial type of
$f$ and polynomial growth condition for $f$.
