On the Refined Esitmates of All Homogeneous Expansions for a Subclass of Biholomorphic Starlike Mappings in Several Complex Variables?

Xiaosong LIU Taishun LIU

Abstract The refined estimates of all homogeneous expansions for a subclass of biholomorphic starlike mappings are mainly established on the unit ball in complex Banach spaces or the unit polydisk in Cn with a unified method. Especially the results are sharp if the above mappings are further k-fold symmetric starlike mappings or k-fold symmetric starlike mappings of order α. The obtained results unify and generalize the corresponding results in some prior literatures.

Keywords Refined estimates of all homogeneous expansions, Starlike mapping, Starlike mapping of order α, k-fold symmetric, Unified method

1 Introduction

In one complex variable of geometric function theory, MacGregor [1] originally established the refined coefficient estimates of biholomorphic starlike functions. Boyd [2] subsequently derived the refined coefficient estimates of starlike functions of orderα. They showed that the above refined estimates of [1] and [2] are sharp if these functions are furtherk-fold symmetric functions. However, the refined coefficient estimates of other subclasses of biholomorphic starlike functions are scarcely discussed. In several complex variables of geometric function theory,Gong [3] posed the profound Bieberbach conjecture in several complex variables, which is that the sharp estimates of all homogeneous expansions for biholomorphic starlike mappings on the unit polydisk in Cnhold. The sharp estimate of the second homogeneous expansion for biholomorphic starlike mappings was proved completely (see [3]). After that Hamada and Honda [4]and Liu and Liu [5] investigated the sharp estimate of the third homogeneous expansion for biholomorphic starlike mappings and starlike mappings of orderαon the unit polydisk in Cnby different methods. In addition, Liu [6] obtained the sharp estimates of all homogeneous expansions for quasi-convex mappings (include quasi-convex mappings of type A and quasi-convex mappings of type B) on the unit polydisk in Cnwith some additional assumptions. Subsequently, Liu and Liu [7] extended the corresponding results of [6] to a general case. Liu, Liu and Xu [8] derived the sharp estimates of all homogeneous expansions for a subclass of biholomorphic starlike mappings in several complex variables as well. With respect to the estimates of homogeneous expansions for a subclass of biholomorphic mappingsfwhich have parametric representation(z=0 is a zero of orderk+1 off(z)?z) on the unit polydisk in Cn, Hamada and Honda[4]and Xu and Liu[9]established the estimates of them(m=k+1,k+2,···,2k)-th homogeneous expansions independently. They both stated that the estimate is only sharp form=k+1. Furthermore,Hamada and Honda[4]investigated the third homogeneous expansion for the above mappings. Recently, Liu and Liu [10] obtained the estimates of all homogeneous expansions for a subclass of biholomorphic mappings which have parametric representation.Many interesting results concerning the estimates of homogeneous expansions may be found in references [11—16].

A natural question arouse great interest of many people: Whether the refined estimates of all homogeneous expansions for a subclasses of biholomorphic starlike mappings which have a concrete parameter representation in several complex variables hold or not? We now provide an affirmative answer partly in this article. That is, we shall establish the refined estimates of all homogeneous expansions for a subclass of biholomorphic starlike mappings which have concrete parametric representation on the unit ball of complex Banach spaces, and also obtain the estimates of all homogeneous expansions for the above generalized mappings on the unit polydisk in Cn.

Throughout this article, we denote byXa complex Banach space with the norm ‖·‖,X?the dual space ofX,Bthe open unit ball inX, andUthe Euclidean open unit disk in C. Also letUndenote the open unit polydisk in Cn, let N+be the set of all positive integers, and let R denote the set of all real numbers. Let the symbol′represent transpose. For eachx∈X{0},

is well defined. We denote byH(B) the set of all holomorphic mappings fromBintoX. It is known that

for allyin some neighborhood ofx∈Biff∈H(B),whereDnf(x)is thenth-Fr′echet derivative offatx, and forn≥1,

We say that a holomorphic mappingf:B→Xis biholomorphic if the inversef?1exists and is holomorphic on the open setf(B). A mappingf∈H(B) is said to be locally biholomorphic if the Fr′echet derivativeDf(x) has a bounded inverse for eachx∈B. Iff:B→Xis a holomorphic mapping, thenfis said to be normalized iff(0) = 0 andDf(0) =I, whereImeans the identity operator fromXintoX.

A normalized biholomorphic mappingf:B→Xis said to be a starlike mapping iff(B) is a starlike domain with respect to the origin.

LetS?(B) be the set of all starlike mappings onB.

We now state the following definitions.

Definition 1.1(see [4])Suppose that g∈H(U)is a biholomorphic function such that g(0) = 1,Reg(ξ)>0, ξ∈U(so, g has real coefficients in its power series expansion), and assume that g satisfies the conditions

It is not difficult to check thatg(ξ) =satisfies the condition of Definition 1.1 for?1 ≤A1

To consider more general cases,we now assume thatg(ξ)=, ξ∈U(A1,A2∈R,|A1|≤1,|A2|≤1).

We denote by Mgthe set

Definition 1.2(see [15])Suppose that f:B→X is a normalized locally biholomorphic mapping. If α∈(0,1)and

then we say that f is a starlike mapping of order α.

LetS?α(B) be the set of all starlike mappings of orderαonB.

Definition 1.3(see [9])Suppose that f:B→X is a normalized locally biholomorphic mapping. If α∈[0,1)and

then we say that f is an almost starlike mapping of order α on B.

We denote byAS?α(B) the set of all almost starlike mappings of orderαonB.

Definition 1.4(see [9])Suppose that f:B→X is a normalized locally biholomorphic mapping. If c∈(0,1)and

then we say that f is a strongly starlike mapping on B.

LetSS?(B) denote the set of all strongly starlike mappings onB.

Definition 1.5(see [17])Let f∈H(B). It is said that f is k-fold symmetric if

for all x∈B, where k∈N+andi=

Definition 1.6(see [18])Suppose thatΩis a domain(connected open set)in X which contains0. It is said that x=0is a zero of order k of f(x)if f(0)=0,···,Dk?1f(0)=0, but Dkf(0)/=0, where k∈N+.

We denote byS?g(B) the subset ofS?(B) consisting of normalized locally biholomorphic mappingsfwhich satisfy (Df(x))?1f(x)∈Mg,andS?g,k+1(B) the subset ofS?g(B) such thatx=0 is a zero of orderk+1 off(x)?x. LetS?k+1(B)(resp.S?α,k+1(B), AS?α,k+1(B), SS?k+1(B))denote the subset ofS?(B) (resp.S?α(B), AS?α(B), SS?(B)) which satisfies thatx= 0 is a zero of orderk+1 off(x)?x.

2 Some Lemmas

In order to establish our main theorems, in this section, it is necessary to provide some lemmas as follows.

Lemma 2.1Suppose that k∈N+, A≥0. Then

ProofIt is readily shown that (2.1) first holds ifq=2. We next assume that

It suffice to prove that (2.1) holds forq=l+1. A direct computation shows that

holds from (2.2). It follows the desired result. This completes the proof.

A direct calculation shows that the following lemma holds (the details are omitted here).

Lemma 2.2Suppose that k∈N+, s=1,2,···, and A≥0. Then

for m≥sk+1.

Lemma 2.3Suppose that k∈N+, f(z) =z+where g(z) =z∈U, A1,A2∈R,|A1|≤1,|A2|≤1. Then

ProofSincethere existsφ∈H(U,U) which statisfies

A straightforward computation shows that

Hence the above relation yields that

Note that

and (2.4). We obtain that

This completes the proof.

Lemma 2.4Suppose that k∈N+, f(z) =z+where g(z) =z∈U, A1,A2∈R,|A1|≤1,|A2|≤1. Then

Especially, if k=1, then

ProofIn view of (2.5), it follows that

This implies that

Similar to the proof of [1, Theorem 1], it yields that

Hence,

Applying an inductive method, we will prove the two following inequalities

and

hold fors=1,2,3,···.

Whens=1, (2.7) holds from (2.3). Also in view of Lemma 2.2 and (2.3), we deduce that

Consequently (2.8) is valid fors= 1 as well. Assume that (2.7) and (2.8) are valid fors=1,2,···,q?1. Lettingp=(q+1)kin (2.6), it yields that

from (2.1). It is shown that (2.7) holds fors=q. On the other hand, whens=q, we prove that

This implies that (2.8) holds fors=q.Hence we derive the desired result from (2.7) readily.This completes the proof.

Remark 2.1Letg1(z) =(α∈(0,1)), g3(z) =(α∈[0,1)), g4(z) =(c∈(0,1)) in Lemma 2.4. Thenf∈S?k+1(U) (S?α,k+1(U),AS?α,k+1(U),SS?k+1(U)), and we get the corresponding results of Lemma 2.4. It is readily shown that the estimates of Lemma 2.4 are sharp iffis ak-fold symmetric starlike function or ak-fold symmetric starlike function of orderα.

Remark 2.2From the proofs of Lemmas 2.3—2.4,it is shown that Lemmas 2.3—2.4 are still valid if the assumptions ofA1,A2∈R are replaced withA1,A2∈C. However, the functionfmust not be a biholomorphic starlike function (even a biholomorphic function).

3 Refined Estimates of All Homogeneous Expansions for a Subclass of Biholomorphic Starlike Mappings in Several Complex Variables

We now present the desired theorems in this section.

Theorem 3.1Let f:B→C ∈H(B), F(x) =xf(x) ∈S?g,k+1(B), g(ξ) =ξ∈U, A1,A2∈R,|A1|≤1,|A2|≤1. Then

In particular, if k=1, then

ProofLetx∈B{0} be fixed, and we denote byx0=Define

It yields that

by a direct calculation, andξ=0 is at least a zero of orderk+1 ofh(ξ)?ξifx=0 is a zero of orderk+1 ofF(x)?x.

On the other hand, we conclude that

from (3.1). Compare the coefficients of the two sides in the above equality. It is shown that

Fork∈N+, we mention that

ifF(x) =xf(x). From Lemma 2.4 and (3.2), it follows the result, as desired. This completes the proof.

Puttingg(ξ)=in Theorem 3.1, then we get the following corollary readily.

Corollary 3.1Let f:B→C ∈H(B), F(x)=xf(x)∈S?k+1(B). Then

The above estimates are sharp for m=sk+1, s=1,2,···. In particular, if k=1, then

The example which shows that the sharpness of estimates of Corollary 3.1 form=sk+1(s=1,2,···) is the same as that of [8, Theorem 2.1].

Setg(ξ) =α∈(0,1) in Theorem 3.1. Then we obtain the following corollary immediately.

Corollary 3.2Let f:B→C ∈H(B), α∈(0,1), F(x)=xf(x)∈S?α,k+1(B). Then

The above estimates are sharp for m=sk+1, s=1,2,···. In particular, when k=1, then

The example which states that the sharpness of estimates of Corollary 3.2 form=sk+1(s=1,2,···) is similar to that of [14, Theorem 2.1].

Settingg(ξ)=, α∈[0,1) in Theorem 3.1, then the following corollary is derived easily.

Corollary 3.3Let f:B→C ∈H(B), α∈(0,1), F(x)=xf(x)∈AS?α,k+1(B). Then

In particular, if k=1, then

Puttingg(ξ)=c∈(0,1)in Theorem 3.1,then the following corollary is given directly.

Corollary 3.4Let f:B→C ∈H(B), c∈(0,1), F(x)=xf(x)∈SS?k+1(B). Then

In particular, when k=1, then

Theorem 3.2Let F(z) = (F1(z),F2(z),···,Fn(z))′∈H(Un), and z= 0is a zero oforder k+1of F(z)?z. Ifz∈Un{0}, where j satisfies the condition|zj|=‖z‖=and g(ξ)=ξ∈U, A1,A2∈R,|A1|≤1,|A2|≤1, then

Especially, if k=1, then

ProofFixz∈Un{0}, and denoteLet

wherejsatisfies the condition|zj|=‖z‖=z∈Un{0},we see that

by a direct calculation. Hence it is shown thathj∈S?g(U), andξ=0 is at least a zero of orderk+1 forhj(ξ)?ξ.

We also show that

from (3.3). It yields that

by comparing the coefficients of the two sides in the above equality. Therefore,it is shown that from Lemma 2.4. In a way similar to that in the proof of[8,Theorem 3.3],we derive the desired result. This completes the proof.

Letg(ξ)=in Theorem 3.2. Then the following corollary is given readily.

Corollary 3.5Let F(z) = (F1(z),F2(z),···,Fn(z))′∈H(Un), and z= 0is a zero oforder k+ 1of F(z)?z. IfReDFj(z)zFj(z)>0, z∈Un{0}, where j satisfies the condition

|zj|=‖z‖= max1≤l≤n|zl|, then

The above estimates are sharp for m=sk+1, s=1,2,···. Especially, when k=1, then

The example which states that the sharpness of estimates of Corollary 3.5 form=sk+1(s=1,2,···) is the same as that of [8, Theorem 3.3].

Puttingg(ξ) =α∈(0,1) in Theorem 3.1, then the following corollary follows immediately.

Corollary 3.6Let α∈(0,1), F(z) = (F1(z),F2(z),···,Fn(z))′∈H(Un), and z= 0is azero of order k+1of F(z)?z. Ifz∈Un{0}, where j satisfies the condition|zj|=‖z‖=then

The above estimates are sharp for m=sk+1, s=1,2,···. In particular, when k=1, then

The example which shows that the sharpness of estimates of Corollary 3.6 form=sk+1(s=1,2,···) is similar to that of [14, Theorem 3.5].

Settingg(ξ)=α∈[0,1) in Theorem 3.2, then the following corollary is derived directly.

Corollary 3.7Let α∈[0,1),F(z) = (F1(z),F2(z),···,Fn(z))′∈H(Un), and z= 0is azero of order k+1of F(z)?z. IfRe>α,z∈Un{0}, where j satisfies the condition|zj|=‖z‖=then

Especially, when k=1, then

Puttingg(ξ)=, c∈(0,1)in Theorem 3.2,then the following corollary is given readily.

Corollary 3.8Let c∈(0,1), F(z) = (F1(z),F2(z),···,Fn(z))′∈H(Un), and z= 0is a zero of order k+1of F(z)?z. Ifz∈Un{0}, where j satisfies thecondition|zj|=‖z‖=then

In particular, when k=1, then

Remark 3.1Theorem 3.1 is the corollary of Theorem 3.2 ifB=Un.

Remark 3.2Corollaries 3.1 and 3.5 are the same as[8, Theorem 2.1]and [8, Theorem 3.3]respectively ifm=sk+1, s=1,2,···.

Remark 3.3Corollaries 3.2 and 3.6 reduce to [14, Theorem 2.1] and [14, Theorem 3.5]respectively ifm=sk+1, s=1,2,···.

According to Theorems 3.1—3.2, we naturally propose the open problem as follows.

Open Problem 3.1LetF(z) ∈ξ∈U, A1,A2∈R,|A1| ≤1,|A2|≤1.Then

The above estimates are sharp forA1= ?1, A2= 1, m=sk+1, s= 1,2,··· andA1=?1, A2=1 ?2α(α∈(0,1)), m=sk+1, s=1,2,···. In particular, ifk=1, then

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