GUO Sheng-ti,XU Qing-hua

(1.College of Mathematics and Information Science,JiangXi Normal,JiangXi Normal University,Nan-Chang 330022,China;2.School of Science,Zhejiang University of Science and Technology,Hangzhou 310023,China)

§1. 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 A consisting of all functions in A which are also univalent in U.Let S?denote the subclass of S consisting of the starlike functions on U.

Let X be a complex Banach space with norm k·k,X?be the dual space of X,B be the unit ball in X,Bnbe the Euclidean unit ball in Cn.Also,let?Undenote the boundary of Un,and?0Unbe the distinguished boundary of Un.

For each x∈X{0},we de fine

According to the Hahn-Banach theorem,T(x)is nonempty.

In[1],Fekete and Szeg? obtained the following classical result.

Let the function f(z)be de fined by(1.1).If f∈S,then

for λ∈[0,1].

The above inequality is known as the Fekete and Szeg? inequality.After that,there were many papers to consider the corresponding problems for various subclasses of the class S,and many interesting results were obtained.For example,in[9],Koepf obtained the following result for S?.

Theorem A(See[9])Let the function f(z)be de fined by(1.1).If f∈S?,then

In contrast,although Fekete and Szeg? inequalities for various subclasses of the class S were established,only a few results are known for the inequalities of homogeneous expansions for subclasses of biholomorphic mappings in several complex variables.Some best-possible results concerning the coefficient estimates for subclasses of holomorphic mappings in several variables were obtained in the works of Graham,Hamada and Kohr[2],Graham et al.[3],Graham et al.[5],Hamada et al.[6],Hamada and Honda[7],Kohr[8],and Xu and Liu[14].

In[15],using some restrictive assumption,Xu and Liu extended Theorem A to the case of the class of starlike mappings de fined on the unit ball in a complex Banach space or on the unit polydisk in Cn.They first obtained the Fekete and Szeg? inequality in several complex variables.Recently,the Fekete and Szeg? inequalities are established for well-known subclasses of holomorphic mappings in several complex variables(see[16-17]).

In this paper,we establish Fekete and Szeg? inequalities for a subclass of starlike mappings de fined on the unit ball in a complex Banach space,on the unit polydisk in Cnor the bounded starlike circular domain in Cn,respectively.These results are natural extensions to higher dimensions of Theorem A.

Let H(B)denote the set of all holomorphic mappings from B into X.It is well known that if f∈H(B),then

for all y in some neighborhood of x ∈ B,where Dnf(x)is the nth-Fréchet derivative of f at x,and for n≥1,

A holomorphic mapping f:B → X is said to be biholomorphic if the inverse f?1exists and is holomorphic on the open set f(B).A mapping f∈H(B)is said to be locally biholomorphic if the Fréchet derivative Df(x)has a bounded inverse for each x ∈ B.If f:B → X is a holomorphic mapping,then f is said to be normalized if f(0)=0 and Df(0)=I,where I represents the linear identity operator from X into X.

Let ? ? Cnbe a bounded starlike circular domain with 0∈ ?,and its Minkowski functional ρ(z)∈ C∞(see Lemma 2.3)except for some lower dimensional manifolds in Cn.The first Fréchet derivative and the m(m>2)-th Fréchet derivative of a mapping f ∈ H(?)at point z ∈ ? are written by Df(z),Dmf(z)(am?1,.),respectively.The matrix representations are

where

Let S?(B)(respectively,be the set of normalized starlike mappings on B(respectively,Un,?).

§2.Some Lemmas

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

L emma 2.1[13]Let f:B→X be a normalized locally biholomorphic mapping.Then f is a starlike mapping on B if and only if

Lemma 2.2Let f:Un→Cnbe a normalized locally biholomorphic mapping.Then f ∈ S?(Un)if and only if

Lemma 2.3(See[10])??Cnis a bounded starlike circular domain if and only if there exists a unique real continuous function ρ :Cn?→ R,called the Minkowski functional of ?,such that

(ii)

(iii)

Furthermore,if ρ(z)(z ∈ ?)∈ C1except for some lower dimensional manifolds in Cn,then the function ρ(z)has the following properties.

Lemma 2.4[10]Let ? ? Cnbe a bounded starlike circular domain with 0 ∈ ?,and its Minkowski functional ρ(z) ∈ C1except for some lower dimensional manifolds in Cn.Letbe a normalized locally biholomorphic mapping.Then f is a starlike mapping on? if and only if

When ?=Bn,obviously,the above inequality is equivalent to the following relation.

§3. Main Results

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

Theorem 3.1LetThen

The above estimate is sharp.

Then g∈H(U),g(0)=1,and

Since F ∈ S?(B),using Lemma 2.1,we obtain

Thus,from Lemma 2.5,we have

Note that F is biholomorphic on B,we get

Using a similar method as in[12](also see[4,Theorem 7.1.14]),we have

Hence,

We easily compute that

In view of(3.4),we have

From this it follows that

Using Taylor series expansions in ξ,we obtain

Comparing the homogeneous expansions of two sides of the above equality,we deduce that

which yields the following

Moreover,from F(x)=xf(x),we have

From(3.6),we conclude that

and

Thus,from(3.2),(3.5),(3.7)and(3.8),we have

Now,we consider the following two cases.

Note that|g0(0)|≤2,we obtain

By combining(3.9)and(3.10),we get the estimate in(3.1),as desired.

To see that the estimation of Theorem 3.1 is sharp,it suffices to consider the following example.

Example 3.1Ifwe consider the following example

Ifwe consider the following example

Using Theorem 1 of[15],we deduce that the mappings F,de fined in(3.11)and(3.12),are in the S?(B).Taking x=ru(0

Remark 3.1When X=C,B=U,Theorem 3.1 is equivalent to Theorem A.

Theorem 3.2LetThen

The above estimation is sharp.

ProofFor z∈Un{0},and denoteLet qj:U→C be given by

Since F ∈ S?(U),from Lemma 2.2,we obtain0,ξ∈ U.Thus,in view of

According to(3.3),we have

or,equivalently,

Using Taylor series expansions in ξ,we obtain

Comparing the homogeneous expansions of two sides of the above equality,we deduce that

On the other hand,fromwe have

Thus,from(3.13),(3.14)and(3.15),we have

Using similar arguments as in the proof of Theorem 3.1,we have

If z0∈ ?0Un,then we obtain

Also since

are holomorphic functions onin view of the maximum modulus theorem of holomorphic functions on the unit polydisc,we obtain

That is

Therefore

as desired.

In order to prove the sharpness,it suffices to consider the following example.

Example 3.2If,we consider the following example

Ifwe consider the following example

From Theorem 3 of[15],we deduce that the mappings F,de fined in(3.16)and(3.17),are in the S?(Un).Taking z=(r,0,···,0)0(0

Remark 3.2When n=1,Theorem 3.2 reduces to Theorem A.

Theorem 3.3LetThen

The above estimation is sharp.

ProofFix z∈ ?{0},and denote.Let p:U ?→ C be given by

Then p∈ H(U),p(0)=1,and since,we deduce that

Using Lemmas 2.4 and 2.5,we obtain

On the other hand,we deduce from De finition 1.3 that

From this it follows that F is biholomorphic on ?,and thusz∈ ?.

With similar reasoning as in the proof of Theorem 3.1,we have

which implies that

In view of(3.20),we have

From this we can conclude that

Using Taylor series expansions in ξ,we obtain

Comparing the homogeneous expansions of two sides of the above equality,we deduce that

That is

Also,sincewe have

From(2.1)and(3.22),we obtain

and

Thus,from Lemma 2.5,(3.21),(3.23)and(3.24),we have

Using similar arguments as in the proof of Theorem 3.1,we obtain

The following Example 3.3 shows that the estimation of Theorem 3.3 is sharp.

where r=sup{|z1|:z=(z1,0,···,0)0∈ ?}.

where r=sup{|z1|:z=(z1,0,···,0)0∈ ?}.

In view of(3.20),we deduce that the mappings F,de fined in(3.25)and(3.26),are in the S?(?).Taking z=Ru(0

Remark 3.3When n=1,?=U,Theorem 3.3 reduces to Theorem A.

When ?=Bn,we immediately obtain the following corollary,which we merely state here without proof.

Corollary 3.1Let.Then

The above estimate is sharp.

Remark 3.4We have just proved that the Fekete and Szeg? inequality for the subclass of starlike mappings in several complex variables.However,the corresponding problem for the general class of starlike mappings remains open.Consequently,we pose the following open problem.

Open ProblemIf F ∈S?(B),then

If X=Cn,B=Un,then

These estimates are sharp.

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