suivant: Final remarks monter: A Bloch Constant for précédent: Bloch constant for -holomorphic

# -holomorphy and quasiregularity

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

on , that is
 (4.3)

Now, , so in this case and . Hence (4.1) becomes

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 -holomorphic
Dominic Rochon
2000-07-26