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

and Applied Analysis 3 By using (16), we deduce that Re x ((DF (x)) −1 F (x))) = Re ( f u (x)) f 󸀠 u (x)) u (x) ‖x‖) > 0 ⇐⇒ Re ( ξf 󸀠 (ξ)


Introduction
Let A be the class of functions of the form which are analytic in the open unit disk We denote by S the subclass of the normalized analytic function class A consisting of all functions which are also univalent in U. Let S * denote the class of starlike functions in U.
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 () =  +  2  2 +  3  3 + ⋅ ⋅ ⋅ ∈ S, then for  ∈ [0, 1].After that, there were many papers to consider the corresponding problems for various subclasses of the class S, and many interesting results were obtained.We choose to recall here the investigations by, for example, Kanas [2] (see also [3][4][5]).
The coefficient estimate problem for the class S, known as the Bieberbach conjecture [6], is settled by de Branges [7], who proved that for a function () =  + ∑ ∞ =2     in the class S, then |  | ≤ , for  = 2, 3, . ... 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.
In contrast, although the coefficient problem for the class S 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]).
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 C  , 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  0   be the distinguished boundary of   .
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  ≥ 1, Furthermore,   () is a bounded symmetric -linear mapping from ∏  =1  into .A holomorphic mapping  :  →  is said to be biholomorphic if the inverse  −1 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 (0) = 0 and (0) = , where  represents the identity operator from  into .Let S() 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 S * () be the set of normalized starlike mappings on .

Some Lemmas
In order to prove the desired results, we first give some lemmas.
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  ∈ S * () if and only if  ∈ S * .This completes the proof of Lemma 4.

Main Results
In this section, we state and prove the main results of our present investigation.
Theorem 1. Suppose  ∈ S * () and The above estimate is sharp.