Let $\mathbb{H}^n$ denote the half-space in
$\mathbb{R}^n$. If $D$ is a domain in $\mathbb{R}^n$, by
$QCH(D)$ we denote the set of Euclidean harmonic quasiconformal
mappings of $D$ onto itself. In Revue Roum. Math.
Pures Appl. Vol. {\bf 51} (2006) 5--6, 711--722, we proved
\begin{thm}\label{th.som}
If $h\in QCH (\mathbb{H}^n)$ and $h(\infty)=
\infty$, then $h$ is euclidean bi-Lipschitz and quasi-isometry
with respect to the $Poincar\acute{e}$ distance.
\end{thm}
We now outline a proof of the theorem: \\
Suppose that $n=3$; the same proof works in general. \\
Let
$h=(h^1, h^2, h^3)$. Since $h^3(x)=x_3$, then $h^3_{x_3}(x)=1$,
and therefore $|h'_{x_3}(x)|\geq 1$. In a similar way,
$|g'_{x_3}(x)|\geq 1$, where $g=h^{-1}$. Hence, for a constant
$c=c(K)$,
$$ |h'(x)|\leq c \quad {\rm and} \quad 1/c \leq l(h'(x)),$$ where
$|h'(x)|= \max\limits_{|h|=1} |h'(x)h| $ and $l(h'(x))=
\min\limits_{|h|=1} |h'(x)h|$; and therefore partial derivatives
of $h$ and $h^{-1}$ are bounded from above; and, in particular,
$h$ is euclidean bi-Lipschitz.
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