Continuing their earlier work on distortion theory, the authors prove some dimension-free distortion theorems for
K
K
-quasiconformal mappings in
R
n
{R^n}
. For example, one of the present results is the following sharp variant of the Schwarz lemma: If
f
f
is a
K
K
-quasiconformal self-mapping of the unit ball
B
n
{B^n}
,
n
⩾
2
n \geqslant 2
, with
f
(
0
)
=
0
f(0) = 0
, then
4
1
−
K
2
|
x
|
K
⩽
|
f
(
x
)
|
⩽
4
1
−
1
/
K
2
|
x
|
1
/
K
{4^{1 - {K^2}}}|x{|^K} \leqslant |f(x)| \leqslant {4^{1 - 1/{K^2}}}|x{|^{1/K}}
for all
x
x
in
B
n
{B^n}
.