The Estimates of All Homogeneous Expansions for a Subclass of ε Quasi-convex Mappings in Several Complex Variables?

Xiaosong LIU Taishun LIU

Abstract The authors obtain the estimates of all homogeneous expansions for a subclass of ε quasi-convex mappings on the unit ball in complex Banach spaces.Moreover,the estimates of all homogeneous expansions for the above generalized mappings on the unit polydisk in Cnare also obtained.Especially,the above estimates are only sharp for a subclass of starlike mappings,quasi-convex mappings and quasi-convex mappings of type A.The results are the generalization of many known results.

Keywords Homogeneous expansion,ε quasi-convex mapping,ε starlike mapping,Starlike mapping,Quasi-convex mapping,Quasi-convex mappings of type A

1 Introduction

It is well known that biholomorphic starlike mappings and biholomorphic convex mappings are two extreme significant families of mappings in the geometric function theory of several complex variables.So the family of ε starlike mappings which were originally introduced by Gong and Liu[2]is a meaningful family of mappings in several complex variables in that it is a family of mappings between the family of biholomorphic starlike mappings and the family of biholomorphic convex mappings.After that Liu and Zhu[5]extended the above family of ε starlike mappings to a new family of mappings which is called ε quasi-convex mappings,and ε∈ [0,1]is widened to ε∈ [?1,1].However,there are only a few results for ε starlike mappings and ε quasi-convex mappings,for instance,the generalized Roper-Suffridge extension operator preserved ε starlikeness on some Reinhardt domains in Cnfor ε∈ [0,1](see[2–3])and ε quasiconvexity on some domains in complex Banach spaces ε∈ [?1,1](see[5]),and the estimates of m-th(m=k+1,k+2,···,2k)homogeneous expansions for ε starlike mappings f(z=0 is a zero of order k+1 of f(z)? z)on the unit polydisk in Cnand ε quasi-convex mappings f(z=0 is a zero of order k+1 of f(z)?z)on the unit ball in complex Banach spaces were established respectively(see[5,7]).

We denote by X the complex Banach space with the norm k·k.Let X?denote the dual space of X,let B be the open unit ball in X,and let U be the Euclidean open unit disk in C.Also,we denote by Unthe open unit polydisk in Cn,and let N?be the set of all positive integers.Let ?Undenote the boundary of Un,(?U)nbe the distinguished boundary of Un.Let the symbol′stand for transpose.For each x∈X{0},we define

Let H(B)be the set of all holomorphic mappings from B into X.It is shown 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 called to be biholomorphic if the inverse f?1exists and is holomorphic on the open set f(B).We say that a mapping f∈H(B)is a locally biholomorphic mapping if the Fréchet 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 if f(0)=0 and Df(0)=I,where I stands for the identity operator from X into X.

A normalized biholomorphic mapping f:B→X is called to be a starlike mapping if f(B)is a starlike domain with respect to the origin.

Now we recall some definitions as follows.

Definition 1.1(see[3])Let f:B→X be a locally biholomorphic mapping with 0∈f(B).f is said to be an ε starlike mapping on B if there exists a positive number ε,0 6 ε 6 1,such that f(B)is starlike with respect to every point in εf(B).

We denote by S?ε(B)the set of all ε starlike mappings on B.

Definition 1.2(see[5])Let ε ∈ [?1,1],and f:B → X be a normalized locally biholomorphic mappings.If

then f is said to be an ε quasi-convex mapping on B.

Let Qε(B)be the set of all ε quasi-convex mappings on B.

It is obviously known that

from Definitions 1.1–1.2.

Definition 1.3(see[9])Suppose that f:B→X is a normalized locally biholomorphic mapping,and denote

then f is said to be a quasi-convex mapping of type A on B.

We denote by QA(B)the set of all quasi-convex mappings of type A on B.

Definition 1.4(see[9])Suppose that f:B→X is a normalized locally biholomorphic mapping.If

then f is said to be a quasi-convex mapping on B.

Let Q(B)be the set of all quasi-convex mappings on B.

Definitions 1.3 and 1.4 are actually the same definitions in one complex variable,and QA(B)=Q(B)(see[9]).

In this paper,we shall establish the estimates of all homogeneous expansions for a subclass of ε quasi-convex mappings on the unit ball in complex Banach spaces.Furthermore,we shall also obtain the estimates of all homogeneous expansions for the above generalized mappings on the unit polydisk in Cn.In particular,the above estimates are only sharp for a subclass of starlike mappings,quasi-convex mappings and quasi-convex mappings of type A.It is shown that a weak version of the Bieberbach conjecture in several complex variables(see[1])will be proved as a corollary,and our results generalize many known results.

2 Estimates of All Homogeneous Expansions for a Subclass of ε Quasiconvex Mappings on the Unit Ball in Complex Banach Spaces

In order to establish the desired theorems in this section,it is necessary to give the lemmas as follows.

Lemma 2.1 Suppose ε∈ [?1,1].If f,g:B → C ∈ H(B),f(0)=g(0)=1,and(1+ ε)(f(x)+Df(x)x)=(f(x)+ εf(?x))g(x),then

Proof In view of the hypothesis of Lemma 2.1,we have

A simple calculation shows that

Compare the homogeneous expansions of the two sides in the above equality.We derived the desired result.This completes the proof.

for l=2,3,···.

Proof It is easily known that(k?1)(1+ε)+(1+(?1)k)ε>0 for ε∈ ??,1?and k=2,3,···.When l=2,a simple calculation shows that the left-hand side of(2.1)isand the right-hand side of(2.1)is also.Hence(2.1)holds for l=2.Assume that(2.1)holds for l=s,namely

When l=s+1,it is shown that

by a direct computation.Hence(2.1)holds for l=s+1.This completes the proof.

Since F(x)=xf(x)∈ Qε(B),it is shown that

from Definition 1.2.Consider

Then g:B →C ∈ H(B),g(0)=f(0)=1,and(1+ε)(f(x)+Df(x)x)=(f(x)+εf(?x))g(x).

We now prove that

by the induction method.When m=2,we deduce that

from Lemma 2.1 and[8,Lemma 2.2].This implies that(2.4)holds for m=2.

Assume that(2.4)holds for m=3,4,···,l.That is,

When m=l+1,by[8,Lemma 2.2]and(2.4),we obtain

This implies that

Hence we see that(2.4)holds for m=l+1.

On the other hand,it is shown that

if F(x)=xf(x).Consequently from(2.4)–(2.5),it follows the result as desired.This completes the proof.

Taking ε=0 in Theorem 2.1,we get the following corollary immediately.

Corollary 2.1 Let f:B →C∈H(B),F(x)=xf(x)∈S?(B).Then

and the above estimates are sharp.

Letting ε=1 in Theorem 2.1,we also obtain the following corollary directly.

Corollary 2.2 Let f:B→C∈H(B),F(x)=xf(x)∈Q(B)(or QA(B)).Then

and the above estimates are sharp.

Remark 2.1 Corollary 2.1 is the same as the case k=1 of[8,Theorem 2.1].

Corollary 2.3 Let ε∈ [0,1],f:B → C∈H(B),F(x)=xf(x)∈ Qε(B).Then

and the above estimates are sharp for ε=0 and ε=1.

Proof It is apparent to see that

from Theorem 2.1.We also know that

In view of the triangle inequality of the norm,it follows the result as desired.This completes the proof.

Remark 2.2 When X=C,B=U,Corollary 2.3 reduces to[6,Theorem 4.1].

Corollary 2.4 Let ε∈ [0,1],f:B → C ∈ H(B),F(x)=xf(x)∈ Qε(B),where B is the unit ball of a complex Hilbert space X.Then

and the above estimates are sharp for ε=0 and ε=1.

Proof It is shown that

(see[4]).Making use of triangle inequalities with respect to the norm in complex Banach spaces,it follows the result as desired.It is not difficult to verify that

and

where kek=1.It yields that

and

by a simple calculation.Put x=re,ξ=Re(0 6 r<1,R>0).Then

Therefore,it is shown that the estimate of Corollary 2.4 is sharp for ε=0 and ε=1.This completes the proof.

3 Sharp Estimates of All Homogeneous Expansions for a Subclass of ε Quasi-convex Mappings on the Unit Polydisk in Cn

Let each mjbe a non-negative integer,N=m1+m2+ ···+mn∈ N?,and mj=0 imply that the corresponding components in Z and F(Z)are omitted.Let Uml(resp.UN)denote the unit polydisk of Cml(l=1,2,···,n)(resp.CN).

Proof Let F(Z)=(F1(Z1),F2(Z2),···,Fn(Zn))′.According to the hypothesis of Theorem 3.1,for any ζ∈,Z=(Z1,Z2,···,Zn)′∈ UN,it yields that

by a simple computation.Note that

if Z=(0,···,Zl,···,0)′∈ UN,l=1,2,···,n.We set

Then it yields that

from Definition 1.2.Taking into account the facts kDmF(0)(Zm)khere kZlkml(resp.kZkN)is written as kZlk(resp.kZk)for simplicity,it is shown that(3.1)holds.This completes the proof.

Setting ε=0 in Theorem 3.1,we get the following corollary readily.

Corollary 3.1 Let fl:Uml→ C ∈ H(Uml),l=1,2,···,n,F(Z)=(Z1f1(Z1),Z2f2(Z2),···,Znfn(Zn))′∈ S?(UN).Then

Taking ε=1 in Theorem 3.1,we also obtain the following corollary immediately.

Corollary 3.2 Let fl:Uml→ C ∈ H(Uml),l=1,2,···,n,F(Z)=(Z1f1(Z1),Z2f2(Z2),···,Znfn(Zn))′∈ Q(UN)(or QA(UN)).Then

Similar to that in the proof of Corollary 2.3,it is not difficult to conclude the following corollary(the details of the proof are omitted here).

Corollary 3.3 Let ε∈ [0,1],fl:Uml→ C ∈ H(Uml),l=1,2,···,n,F(Z)=(Z1f1(Z1),Z2f2(Z2),···,Znfn(Zn))′∈ Qε(UN).Then

and the above estimates are sharp for ε=0 and ε=1.

Proof Fix z∈Un{0},and set z0=.Let

where j satisfies the conditionIt yields that

by a direct calculation.Therefore it is shown that

On the other hand,it is readily shown that

from(3.2).Comparing the coefficients of the two sides in the above equality,it yields that

Consequently by Theorem 2.1(the case of X=C,B=U),we have

When z0∈ (?U)n,we have

Also noticing that DmFl(0)(zm)is a holomorphic function on Un,in view of the maximum modulus theorem of holomorphic functions on the unit polydisk,we conclude that

This implies that

Therefore it follows that

This completes the proof.

Putting ε=0 in Theorem 3.2,we obtain the following corollary directly.

Corollary 3.4 Suppose F(z)=(F1(z),F2(z),···,Fn(z))′∈ H(Un).z∈Un{0},where j satisfies the conditionthen

Letting ε=1 in Theorem 3.2,we also get the following corollary easily.

Corollary 3.5 Suppose F(z)=(F1(z),F2(z),···,Fn(z))′∈ H(Un).If Re0,ζ∈U,z∈Un{0},where j satisfies the conditionthen

With the analogous arguments as in the proof of Corollary 2.3,it is easy to deduce the following corollary(the details of the proof are omitted here).

Corollary 3.6 Suppose that ε∈ [0,1],F(z)=(F1(z),F2(z),···,Fn(z))′∈ H(Un).Ifζ∈ U,z ∈ Un{0},where j satisfies the condition|zj|=kzk=then

and the above estimates are sharp for ε=0 and ε=1.

Remark 3.1 We readily see that Theorem 2.1 is the special case of Theorem 3.2 when X=Cn,B=Un,and Theorem 3.1 is the special case of Theorem 3.2 for m1=n,ml=0,l=2,···,n or ml=1,l=1,2,···,n as well.

Remark 3.2 For the estimates of all homogeneous expansions for a subclass of ε quasiconvex mappings,ε∈ ??13,1?is a sufficient condition in Theorems 2.1,3.1–3.2.We do not know the corresponding results for the case ε∈ ??1,?13?nowadays.However,concerning the growth theorem and the upper bounds of the distortion theorem for a subclass of ε quasi-convex mappings,ε∈ [0,1]seems to be a necessary condition in Corollaries 2.3–2.4,3.3 and 3.6.This implies that the condition of growth theorem and the upper bounds of the distortion theorem for a subclass of ε quasi-convex mappings is stronger than the condition of the estimates of all homogeneous expansions for a subclass of ε quasi-convex mappings.

AcknowledgementThe authors would like to thank the referees for their useful comments.