Abstract and Applied Analysis

Volume 2014 (2014), Article ID 807026, 6 pages

http://dx.doi.org/10.1155/2014/807026

Research Article

On the Fekete and Szegö Problem for the Class of Starlike Mappings in Several Complex Variables

Qing-Hua Xu¹ and Tai-Shun Liu²

¹College of Mathematics and Information Science, Jiangxi Normal University, Nanchang 330022, China

²Department of Mathematics, Huzhou Teacher’s College, Huzhou 313000, China

Received 22 May 2014; Accepted 7 August 2014; Published 3 September 2014

Academic Editor: Carlos Vazquez

Copyright © 2014 Qing-Hua Xu and Tai-Shun Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

Remark 4. When , Theorem 3 reduces to Theorem A.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by NNSF of China (Grant no. 11261022), the Jiangxi Provincial Natural Science Foundation of China (Grant no. 20132BAB201004), the Natural Science Foundation of Department of Education of Jiangxi Province, China (Grant no. GJJ12177), and the Zhejiang Provincial Natural Science Foundation of China (Grant no. Y6110053).

References

M. Fekete and G. Szegö, “Eine bemerkunguber ungerade schlichte Funktionen,” Journal London Mathematical Society, vol. 8, pp. 85–89, 1933.

S. Kanas, “An unified approach to the Fekete-Szegö problem,” Applied Mathematics and Computation, vol. 218, no. 17, pp. 8453–8461, 2012.

A. Pfluger, “The Fekete-Szegö inequality for complex parameters,” Complex Variables: Theory and Application, vol. 7, no. 1–3, pp. 149–160, 1986.

R. R. London, “Fekete-Szegö inequalities for close-to-convex functions,” Proceedings of the American Mathematical Society, vol. 117, no. 4, pp. 947–950, 1993.

B. Bhowmik, S. Ponnusamy, and K. Wirths, “On the Fekete–Szegö problem for concave univalent functions,” Journal of Mathematical Analysis and Applications, vol. 373, no. 2, pp. 432–438, 2011.

L. Bieberbach, “Über die Koeffizienten der einigen Potenzreihen welche eine schlichte Abbildung des Einheitskreises vermitten,” S. B. Preuss. Akad. Wiss., vol. 38, pp. 940–955, 1916.

L. de Branges, “A proof of the Bieberbach conjecture,” Acta Mathematica, vol. 154, no. 1-2, pp. 137–152, 1985.

H. Cartan and P. Montel, “Sur la possibilité d’étendre aux fonctions de plusieurs variables complexes la théorie des fonctions univalentes,” in Lecons sur les Fonctions Univalentes ou Multivalentes, Gauthier-Villars, Paris, France, 1933.

S. Gong, The Bieberbach Conjecture, American Mathematical Society, International Press, Providence, RI, USA.

I. Graham, H. Hamada, and G. Kohr, “Parametric representation of univalent mappings in several complex variables,” Canadian Journal of Mathematics, vol. 54, no. 2, pp. 324–351, 2002.

I. Graham, G. Kohr, and M. Kohr, “Loewner chains and parametric representation in several complex variables,” Journal of Mathematical Analysis and Applications, vol. 281, no. 2, pp. 425–438, 2003.

H. Hamada, T. Honda, and G. Kohr, “Growth theorems and coefficients bounds for univalent holomorphic mappings which have parametric representation,” Journal of Mathematical Analysis and Applications, vol. 317, no. 1, pp. 302–319, 2006.

H. Hamada and T. Honda, “Sharp growth theorems and coefficient bounds for starlike mappings in several complex variables,” Chinese Annals of Mathematics B, vol. 29, no. 4, pp. 353–368, 2008.

G. Kohr, “On some best bounds for coefficients of several subclasses of biholomorphic mappings in Cn,” Complex Variables, vol. 36, pp. 261–284, 1998.

X. Liu and T. Liu, “The sharp estimates of all homogeneous expansions for a class of quasi-convex mappings on the unit polydisk in ,” Chinese Annals of Mathematics B, vol. 32, no. 2, pp. 241–252, 2011.

Q. H. Xu and T. S. Liu, “On coefficient estimates for a class of holomorphic mappings,” Science in China A: Mathematics, vol. 52, no. 4, pp. 677–686, 2009.

W. Koepf, “On the Fekete-Szegö problem for close-to-convex functions,” Proceedings of the American Mathematical Society, vol. 101, no. 1, pp. 89–95, 1987.

T. J. Suffridge, “Starlike and convex maps in Banach spaces,” Pacific Journal of Mathematics, vol. 46, pp. 575–589, 1973.

I. Graham and G. Kohr, Geometric Function Theory in One and Higher Dimensions, Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, Inc., New York, 2003.