SHARP DISTORTION THEOREMS FOR A CLASSOF BIHOLOMORPHIC MAPPINGS IN SEVERAL COMPLEX VARIABLES*

Xiaosong LIU (刘小松)

School of Mathematics and Statistics,Lingnan Normal University，Zhanjiang 524048,China E-mail: laszhinc@163.com

Moreover, the above estimates are sharp.

It should be noted, however, that Cartan [3] provided a counterexample to state that the above theorems are invalid in several complex variables. This is important, given the significance of the growth and distortion theorems and the Goluzin type distortion theorem for biholomorphic functions in one complex variable. We attempt to study the growth and distortion theorems and the Goluzin type distortion theorem for biholomorphic mappings defined on the unit ball of complex Banach spaces and the unit polydisk in Cnunder certain additional assumptions. Concerning the growth and distortion theorems for subclasses of biholomorphic mappings,significant results were first established by Barnard,FitzGerald and Gong[1,2],who dealt with the growth theorem, covering theorems for biholomorphic starlike mappings in Cn,and the distortion theorem for biholomorphic convex mappings on the Eucliden unit ball in C2. After that, Gong and Liu [5], and Hammada and Kohr [9] established the distortion theorem for biholomorphic convex mappings in finite and infinite dimensional spaces, respectively.More meaningful results concerning the sharp growth and distortion theorems for subclasses of biholomorphic mappings are found in references [6-17].

In this paper, we denote by X a complex Banach space with the norm ‖ . ‖, and by X∗the dual space of X. Let B denote the open unit ball in X, and let U be the Euclidean open unit disk in C. At the same time, let Unbe the open unit polydisk in Cn, let N be the set of all nonnegative integers, and let N∗be the set of all positive integers. We denote by ∂Unthe boundary of Un, by (∂U)nthe distinguished boundary of Un, and by Unthe closure of Un.Let the symbol＇mean transpose. For arbitrary x ∈X{0}, we denote that

We say that a mapping f ∈H(B,X) is locally biholomorphic if the Fr´echet derivative Df(x)has a bounded inverse for each x ∈B. If f :B →X is a holomorphic mapping,then we say that f is normalized when f(0) = 0 and Df(0) = I, where I represents the identity operator from X into X. It is shown that a holomorphic mapping f :B →X is biholomorphic if the inverse f-1exists and is holomorphic on the open set f(B). Let S(B) be the set of all biholomorphic mappings which are defined on B. A normalized biholomorphic mapping f : B →X is said to be a starlike mapping if f(B) is a starlike domain with respect to the origin. We denote by S∗(B) the set of all biholomorphic starlike mappings defined on B.

2 The Sharp Growth Theorem and the Distortion Theorem of the Frech´et Derivative for a Class of Biholomorphic Mappings in Several Complex Variables

Therefore, in view of (2.5)-(2.8), we obtain the desired result. With arguments analogous to those in the proof of [14, Theorem 2.4], we show that the estimates of Theorem 2.4 are sharp.This completes the proof. □

Remark 2.5 The second inequalities of Theorems 2.2-2.4 include [14, Corollaries 2.1,2,3, 2.4], due to S∗(B)⊂S(B).

3 The Sharp Distortion Theorem of a Jacobi Determinant for a Class of Biholomorphic Mappings in Cn

In this section, we denote by JF(z) the Jacobi matrix of the holomorphic mapping F(z),and by det JF(z) the Jacobi determinant of the holomorphic mapping F(z). We also let B denote the unit ball of Cnwith the arbitrary norm ‖.‖, and let Indenote the unit matrix of Cn.

Remark 3.3 Theorems 3.1-3.2 include[14, Corollaries 3.1, 3.4](the case k = 1), due to S∗(B)⊂S(B).

4 A Sharp Goluzin Type Distortion Theorem for a Class of Biholomorphic Mappings in Several Complex Variables

In order to establish the desired results in this section, we need to provide the following lemma:

Combining (4.1) and (4.2), the desired result follows. This completes the proof. □

We now present the desired theorem in this section.

Theorem 4.2 Let f :B →C ∈H(B), F(x)=xf(x)∈S(B). Then

Hence we show that the estimates of Theorem 4.4 are sharp. This completes the proof. □

Consequently,the desired result follows from(4.9)and(4.10). The sharpness of the estimates of Theorem 4.6 are similar to those in the proof of Theorem 4.5(the case ml=1(l=,1,2,...,n)).This completes the proof. □