New Subclasses of Biholomorphic Mappings and the Modi fi ed Roper-Suffridge Operator∗

Chaojun WANGYanyan CUIHao LIU

1 Introduction

In 1995,the Roper-Suffridge operator

was introduced in[17],where f(z1)is a univalent holomorphic function on the unit disk D,and z=(z1,z0) ∈ Bn,z1∈ D,z0=(z2,···,zn) ∈ Cn−1,By the Roper-Suffridge operator,we can construct a normalized locally biholomorphic mapping on Bnthrough a normalized locally biholomorphic function on D.Therefore,the Roper-Suffridge operator provides a powerful tool for constructing biholomorphic mappings with special geometric properties in several complex variables.By far,there have been lots of beautiful results about the generalized Roper-Suffridge operator(see[2,4–5,7–12,18]).

In 2005,Muir and Suffridge[14]extended the Roper-Suffridge operator to be

where f is a normalized biholomorphic function on D,z=(z1,z0)∈Bn,z1∈D,z0=and P:Cn−1→ C is a homogeneous polynomial of degree 2.Muir and Suffridge proved that the operator(1.1)preserves starlikeness and convexity onrespectively.By(1.1),Muir and Suffridge[15–16]discussed the extremal mapping of convex mapping on Bn.

Wang and Liu[20]modified the operator(1.1)to be

where P:Cn−1→ C is a homogeneous polynomial of degree m.They showed that the operator(1.2)preserves almost starlikeness of order α ∈ [0,1)and starlikeness of order α ∈ (0,1)under some conditions for P.In 2011,Feng and Yu[3]showed that the operator(1.2)preserves almost spirallikeness of type β and order α,spirallikeness of type β and order α,and strong spirallikeness of type β and order α on Bn.

In 2008,Muir[13]introduced the following generalized Roper-Suffridge operator on the unit ball in complex Banach spaces:

where z=(z1,),f(z1)is a normalized univalent holomorphic function on D,and G is a holomorphic function in Cn−1with G(0)=0,DG(0)=0, γ ≥ 0 and[f′(z1)]β|z1=0=1.Muir discussed the relationship between[ΦG,β(f)](z)and Loewner chains.

Now,we extend the Roper-Suffridge operator to be

In this paper,we mainly seek conditions for G under which the modified operator[ΦG,γ(f)](z)preserves the properties of subclasses of biholomorphic mappings.In Sections 2–3,by the properties of k-fold symmetric mappings and the growth theorems of subclasses of biholomorphic mappings,we study the properties of modified operator[ΦG,γ(f)](z)for strong and almost spirallike mappings of type β and order α on Bn.Thus we obtain that[ΦG,γ(f)](z)preserves starlikeness of order α,strong starlikeness of order α,strong and almost starlikeness of order α,and strong spirallikeness of type β.

In order to get the main results,we need the following definitions and lemma.

Definition 1.1(see[19])Let Ω ⊂ Cnbe a bounded starlike circular domain with 0 ∈ Ω,and the Minkowski functional ρ(z)of Ω be C1except for some submanifolds of lower dimensions.Let f(z)be a normalized locally biholomorphic mapping on Ω,and let−1≤ A ＜ B ＜ 1,

Then we call f(z)∈(A,B).

For Ω=D,the condition reduces to

If A= −1 and B=1−2α,then f(z)is a starlike function of order α.

If A= −α and B= α,then f(z)is a strong starlike function of order α.

Definition 1.2(see[1])Let f(z)be a normalized locally biholomorphic mapping on Bn.Let α ∈ [0,1),and

Then f(z)is called a strong and almost spirallike mapping of type β and order α on Bn.

For Ω=D,the condition reduces to

Setting α =0, β =0 and α = β =0,Definition 1.2 reduces to the definition of strong spirallike mappings of type β,strong and almost starlike mappings of order α,and strong starlike mappings,respectively.

Lemma 1.1(see[13])Let P(z)be a homogeneous polynomial of degree m,and let DP(z)be the Fr´echet derivative of P at z.Then

2 The Invariance of (A,B)

Lemma 2.1 Let

where z=(z1,z0),f(z1)is a normalized univalent holomorphic function on D,G is holomorphic in Cn−1with G(0)=0,DG(0)=0,γ≥ 0 andThe homogeneous expansionwhere Pj(z)is a homogeneous polynomial of degree j.Then

Proof Since

we get

where In−1is the identity operator on Cn−1.

Let(DF(z))−1F(z)=h(z).Then DF(z)h(z)=F(z).Let h(z)=(A,B)′.From Lemma 1.1,we have

It follows that

So we have the desired conclusion.

Lemma 2.2 Let f(z1)∈(A,B)with f′(z1)0 and−1≤A＜B≤Let

Then

Proof Since f(z1)∈(A,B),by Definition 1.1,we have

Thus|q(z1)|＜Let

Thenfor z10.It follows thatTherefore

Then

For B≤0,we have

For 0＜B≤,we have

Hence|g(z1)|＜1 for−1≤A＜B≤Observing that g(z1)is holomorphic on D and g(0)=0,from the Schwarz lemma,we obtain|g(z1)|≤|z1|,that is

In the following,f(z)is said to be k-fold symmetric(see[10]),where k is a positive integer if

Lemma 2.3(see[19])Let Ω⊂Cnbe a bounded starlike circular domain,and the Minkowski functional ρ(z)of Ω be C1except for some submanifolds of lower dimensions.Let f(z)∈(A,B)be k-fold symmetric.Then

or equivalently,

The above estimates are all accurate.

The following are our main results.

Theorem 2.1 Let f(z1)∈(A,B)be a k-fold symmetric function with f′(z1)0,−1≤A＜B≤and k is a positive integer.Let F(z)be the mapping defined in Lemma 2.1 with γ∈[0,1).

(1)If A=0,then F(z)∈(A,B)provided that

(2)If 0＜A＜1,then F(z)∈(A,B)provided that Pj=and

(3)If−1≤A＜0,then F(z)∈S∗Ω(A,B)provided that γ＜min

Proof By Definition 1.1,we only need to prove

It is obvious that(2.1)holds for z0=0,since F(z)is holomorphic for z∈Bn(z00).From the maximum modulus principle of holomorphic functions,we only need to prove that(2.1)holds for z ∈ ∂Bn(z00).In the following,let ‖z‖2=|z1|2+ ‖z0‖2=1.

Since f(z1)∈S∗Ω(A,B),by Definition 1.1,we have

Let

It follows that

From Lemma 2.1,we have

Since f(z1)∈(A,B)is k-fold symmetric,by Lemma 2.3,we have

We discuss the properties of F(z)from Lemma 2.2 in the following 3 cases.(1)In the case of A=0,we have

where

Then F(z)∈S∗Ω(A,B)by Definition 1.1.

(2)In the case of 0＜A＜1,it is obvious that 1−A|z1|k≥1−|z1|.Then

for A＜B.Therefore

where

Hence,F(z)∈(A,B)by Definition 1.1.

(3)In the case of−1≤A＜0,it is obvious that

Furthermore,for A＜B.Then

Similar to the case of 0＜A＜1,we obtain

whereand

Hence,F(z)∈(A,B)by Definition 1.1.

Setting A=−(B+2α)=−1 and A=−B=−α in Theorem 2.1,respectively,we can get the following results.

Corollary 2.1 Let f(z1)be a starlike function of order α on D with f′(z1)0, α ∈ ?,1）,and let f(z1)be k-fold symmetric.Let F(z)be the mapping defined in Lemma 2.1 with γ∈[0,1),and let Pj=0 forand

Then F(z)is a starlike mapping of order α on Bn.

Corollary 2.2 Let f(z1)be a strong starlike function of order α on D with f′(z1)0, α ∈（0,?,and let f(z1)be k-fold symmetric.Let F(z)be the mapping defined in Lemma 2.1 withand let Pj=0 forand

Then F(z)is a strong starlike mapping of order α on Bn.

Setting γ=j=m,we can get the following result.

Corollary 2.3 Let f(z1)∈(A,B)be a k-fold symmetric function with f′(z1)0,and k is a positive integer.Let

where z=(z1,z0)∈ Bn,P(z0)is a homogeneous polynomial of degree m in Cn−1with m ∈Z+,m≥2 and

(1)If A=0,then F(z)∈(A,B)provided that

(2)If 0＜A＜1,then F(z)∈(A,B)provided that mand

(3)If−1≤A＜0,then F(z)∈(A,B)provided that mand

Remark 2.1 Setting k=1 in Theorem 2.1 and Corollaries 2.1–2.3,we have the corresponding simplified results.

3 The Invariance of Strong and Almost Spirallike Mappings of Type β and Order α

We begin with some helpful lemmas.

Lemma 3.1 Let f(z1)be a strong and almost spirallike function of type β and order α on D with f′(z1)0,α ∈ [0,1),and c ∈ (0,].Let

Then

Proof Since f(z1)is a strong and almost spirallike function of type β and order α on D,by Definition 1.2,we have

That isLet

Thenfor z10.It follows thatTherefore

Then

for c∈(0,].Observing that g(z1)is holomorphic on D and g(0)=0,we have|g(z1)|≤|z1|by the Schwarz lemma.Thus

Lemma 3.2(see[6])Let f(z)be a normalized univalent holomorphic function on D.Then

The following are our main results.

Theorem 3.1 Let f(z1)be a strong and almost spirallike function of type β and order α on D with f′(z1)0,α ∈ [0,1),andLet F(z)be the mapping defined in Lemma 2.1 with γ ＜Pj=0and

Then F(z)is a strong and almost spirallike mapping of type β and order α on Bn.

Proof By Definition 1.2,we only need to prove

It is obvious that(3.1)holds for z0=0,since F(z)is holomorphic for z∈Bn(z00).From the maximum modulus principle of holomorphic functions,we only need to prove that(3.1)holds for z ∈ ∂Bn(z00).In the following,let ‖z‖2=|z1|2+ ‖z0‖2=1.

Since f(z1)is a strong and almost spirallike function of type β and order α on D,by Definition 1.2,we have

Let

It follows that

By Lemma 2.1,we obtain

From Lemma 3.1,we have

Furthermore,From the condition

and Lemma 3.2,we get

Hence F(z)is a strong and almost spirallike mapping of type β and order α on Bn.

Setting γ=and j=m in Theorem 3.1,we get the following result.

Corollary 3.1 Let f(z1)be a strong and almost spirallike function of type β and order α on D with f′(z1)0,α ∈ [0,1), β ∈ （−）and c ∈ （0,?,and let P(z0)be a homogeneous polynomial of degree m in Cn−1,where m∈Z+,m≥6.Let

where z=(z1,z0)∈Bn,and the branch of the power function is chosen such that=1.If then F(z)is a strong and almost spirallike mapping of type β and order α on Bn.

Remark 3.1 Setting α =0,β =0 in Theorem 3.1 or Corollary 3.1 respectively,we get the corresponding results for strong spirallike mappings of type β,and strong and almost starlike mappings of order α on Bn.

AcknowledgementThe authors are grateful to the anonymous referees for their valuable comments and suggestions which helped to improve the quality of the paper.

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