Abstract

Let be the familiar class of normalized univalent functions in the unit disk. Fekete and Szegö proved the well-known result for . We investigate the corresponding problem for the class of starlike mappings defined on the unit ball in a complex Banach space or on the unit polydisk in , which satisfies a certain condition.

1. Introduction

Let be the class of functions of the form which are analytic in the open unit disk We denote by the subclass of the normalized analytic function class consisting of all functions which are also univalent in . Let denote the class of starlike functions in .

It is well known that the Fekete and Szegö inequality is an inequality for the coefficients of univalent analytic functions found by Fekete and Szegö [1], related to the Bieberbach conjecture. Finding similar estimates for other classes of functions is called the Fekete and Szegö problem.

The Fekete and Szegö inequality states that if , then for . After that, there were many papers to consider the corresponding problems for various subclasses of the class , and many interesting results were obtained. We choose to recall here the investigations by, for example, Kanas [2] (see also [3–5]).

The coefficient estimate problem for the class , known as the Bieberbach conjecture [6], is settled by de Branges [7], who proved that for a function in the class , then , for .

However, Cartan [8] stated that the Bieberbach conjecture does not hold in several complex variables. Therefore, it is necessary to require some additional properties of mappings of a family in order to obtain some positive results, for instance, the convexity and the starlikeness.

In [9], Gong has posed the following conjecture.

Conjecture A. If is a normalized biholomorphic starlike mapping, where is the open unit polydisk in , then

In contrast, although the coefficient problem for the class had been completely solved, only a few results are known for the inequalities of homogeneous expansions for subclasses of biholomorphic mappings in several complex variables (see, for detail, [9]).

Recently, some best-possible results concerning the coefficient estimates for subclasses of holomorphic mappings in several variables were obtained in work of Graham et al. [10], Graham et al. [11], Hamada et al. [12], Hamada and Honda [13], Kohr [14], X. Liu and T. Liu [15], and Xu and Liu [16].

In [17], Koepf obtained the following result for .

Theorem A. Let . Then The above estimation is sharp.

It is natural to ask whether we can extend Theorem A to higher dimensions.

In this paper, we will establish inequalities between the second and third coefficients of homogeneous expansions for starlike mappings defined on the unit ball in Banach complex spaces and the unit polydisc in , respectively, which are the natural extension of Theorem A to higher dimensions.

Let be a complex Banach space with norm ; let be the dual space of ; let be the unit ball in . Also, let denote the boundary of , and let be the distinguished boundary of .

For each , we define According to the Hahn-Banach theorem, is nonempty.

Let denote the set of all holomorphic mappings from into . It is well known that if , then for all in some neighborhood of , where is the th-Fréchet derivative of at , and, for , Furthermore, is a bounded symmetric -linear mapping from into .

A holomorphic mapping is said to be biholomorphic if the inverse exists and is holomorphic on the open set . A mapping is said to be locally biholomorphic if the Fréchet derivative has a bounded inverse for each . If is a holomorphic mapping, then is said to be normalized if and , where represents the identity operator from into . Let be the set of all normalized biholomorphic mappings on . We say that is starlike if is biholomorphic on and is starlike with respect to the origin. Let be the set of normalized starlike mappings on .

Suppose that is a bounded circular domain. The first Fréchet derivative and the -th Fréchet derivative of a mapping at point are written by , , respectively. The matrix representations are where , .

2. Some Lemmas

In order to prove the desired results, we first give some lemmas.

Lemma 1 (see [18]). Let be a normalized locally biholomorphic mapping. Then is a starlike mapping on if and only if

Lemma 2. Let be a normalized locally biholomorphic mapping. Then if and only if where and .

Lemma 3 (see [19]). Let , and , ; then

Lemma 4. Suppose that . Then defined by , where , belongs to if and only if .

Proof. Denote ; since , we have Straightforward calculation yields It is not difficult to check that Hence By using (16), we deduce that Therefore, by Lemma 1, we obtain that if and only if . This completes the proof of Lemma 4.

3. Main Results

In this section, we state and prove the main results of our present investigation.

Theorem 1. Suppose and Then The above estimate is sharp.

Proof. Fix and denote . Let be given by where . Then , , and Since , from Lemma 1, we have In view of Lemma 3, we obtain that That is,
On the other hand, since , we have Comparing with the homogeneous expansion of two sides of the above equality, we obtain Equation (27) may be rewritten as follows: Thus, from (18) of Theorem 1, (24), (26), and (28), we deduce that If now , then On the other hand, if , then we use and get The following example shows that the estimation of Theorem 1 is sharp.
Example. If , we consider the following example: By Lemma 4, we obtain that .
It is not difficult to check that the mapping satisfies the condition of Theorem 1. Setting () in (32), we obtain that If , we consider the following example: In view of Lemma 4, we deduce that .
It is not difficult to verify that the mapping satisfies the condition of Theorem 1. Taking () in (34), we have This completes the proof of Theorem 1.

Remark 2. When , , Theorem 1 is equivalent to Theorem A.

Theorem 3. Suppose and for , where , . Then The above estimate is sharp.

Proof. For any , denote . Let be given by where and satisfies . Then , , and Since , from Lemma 2, we deduce that , . Therefore, according to Lemma 3, we have Hence, in view of (26), (28), and (32) of Theorem 3, we obtain that If now , then On the other hand, if , then we use and get Then, by using (42) and (43), we have If , then we have Also since is a holomorphic function on , in view of the maximum modulus theorem of holomorphic function on the unit polydisc, we obtain That is, Hence
Finally, in order to see that the estimation of Theorem 3 is sharp, it suffices to consider the following mappings.
If , we consider the following example:
If , we consider the following example: In view of Problem  6.2.5 of [19], we deduce that the mappings , defined in (50) and (51), are in the class .
It is not difficult to verify that the mappings defined in (50) and (51) satisfy the condition of Theorem 3. Taking () in (50) and (51), respectively, we deduce that the equality in (37) holds true. This completes the proof of Theorem 3.