As we saw in the introduction, there is a Bloch theorem for
K-quasiregular mappings. Then, to justify our Bloch theorem on the unit ball, we
need to prove that the new class of mappings is not totally included in
the class of -quasiregular mappings.
First, with Definition 1 it is easy to show the
following characterization:
Remark 1
If
then is -quasiregular iff
The following examples will clearly show that a
-holomorphic mapping is not necessarily quasiregular.
Example 1
Let , where and are in
. If
is noninvertible then, f is not quasiregular. If is
invertible, then is -quasiregular for
where In particular, if
, then
is conformal. However, we note in this example, that
is a linear transformation which is nondegenerate if and only if
is invertible.
In fact, any injective holomorphic mapping of the closed ball into
is -quasiregular for some , but it is interesting to estimate .
In the last example, it is important to specify that is
clearly an entire -holomorphic mapping and that the
multiplication is the multiplication between bicomplex numbers.
The next examples will show that
there exist some nontrivial mappings which are simultaneously quasiregular and
-holomorphic with
.
Example 2
If then is -quasiregular on
iff
. Moreover, because
,
we have
.
Proof.
Let and be holomorphic functions on . Then we know
that
is
-holomorphic with
. First, we seek conditions on and
such that is -quasiregular. By Remark 1 it is easy to show that
is -quasiregular on iff
Let
. Then, because
, we have that . Moreover,
and then
, i.e.
.
In Example 2, because , we know by Theorem 6
that
. Since the mapping is
-quasiregular, we already know that
by Theorem
2. However, Theorems 2 and 6 merely
assert the existence of the constants and without giving
any estimates for these constants. From Theorem 9, on the other hand, we
have an interesting lower estimate
, by invoking lower
estimates on the classical Bloch constant [6]. One
can also give lower estimates for the Bloch constant for
-quasiregular holomorphic mappings [7].
The next example is a mapping for which
by Theorem 6, but for which it is impossible to invoke
Theorem 2.
Example 3
If
, then is an entire -holomorphic (normalized)
mapping, but for all is not -quasiregular.
Proof.
The function is normalized because
and then . Also,
is in with
which is noninvertible.
Hence cannot satisfy the criteria of Remark 1 at and then for all , cannot be
-quasiregular. Actually, quasiregular holomorphic mappings in
are necessarily
locally injective hence locally quasiconformal, but we wished
to avoid invoking this rather deep theorem (see [8]).
Now, we show that our class of mappings includes some -quasiregular mappings
for arbitrary values of K. Then, Theorems 6 and 9 give geometric
information about -quasiregular mappings for a subclass including different
values of .
Example 4
If
,
then is an entire -holomorphic (normalized)
mapping which is -quasiregular in with becoming
necessarily bigger as increases.
Proof.
First, we see that is normalized because
and
i.e.
. Also, by (4.1)
is -quasiregular on iff
on . Then, we must have
However,
, where
can be taken to be positive. Finally we see that the last expression
goes to infinity as
.
Finally, we give an example of a mapping which is -holomorphic
and biholomorphic without being quasiregular on the unit ball of
. In fact, we show
that the class of -biholomorphic mappings cannot be totaly included
in the class of quasiconformal mappings on the unit ball.
Example 5
If
then is an entire -holomorphic (normalized) mapping
which is biholomorphic but not quasiregular on the unit ball of
.
Proof.
Because is nonzero on
for and
, then is -biholomorphic
on . However, cannot be K-quasiregular on because
the relationship (4.1) will fail for all as
with
. suivant:Final remarks monter:A Bloch Constant for précédent:Bloch constant for -holomorphicDominic Rochon 2000-07-26