Mappings of finite distortion of polynomial type

Suppose that $f:{\text{bR}}^n\to {\text{bR}}^n$ 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 $N(f,{\text{bR}}^n)$, polynomial type of $f$ and polynomial growth condition for $f$.