SUBCLASSES OF BIHOLOMORPHIC MAPPINGS UNDER THE EXTENSION OPERATORS?

Chaojun WANG

College of Mathematics and Statistics,Zhoukou Normal University,Zhoukou 466001,China,

E-mail:wang9907081@163.com

Yanyan CUI

College of Mathematics and Statistics,Zhoukou Normal University,Zhoukou 466001,China;College of Mathematics and Information Science,Hebei Normal University,Shijiazhuang 050016,China

E-mail:cui9907081@163.com

Hao LIU

Institute of Contemporary Mathematics,Henan University,Kaifeng 475001,China

E-mail:haoliu@henu.edu.cn

Abstract In this article,we mainly study the invariance of some biholomorphic mappings with special geometric characteristics under the extension operators.First we generalize the Roper-Su ff ridge extension operators on Bergman-Hartogs domains.Then,by the geometric characteristics of subclasses of biholomorphic mappings,we conclude that the modi fi ed Roper-Su ff ridge operators preserve the properties of(β,A,B),parabolic and spirallike mappings of type β and order ρ,strong and almost spirallike mappings of type β and order α as well as almost starlike mappings of complex order λ onunder di ff erent conditions,respectively.The conclusions provide new approaches to construct these biholomorphic mappings in several complex variables.

Key words spirallike mappings;Roper-Su ff ridge operator;Bergman-Hartogs domains

1 Introduction

The constructing of biholomorphic mappings with particular geometric properties plays an important role in several complex variables.In recent years,there were lots of subclasses of biholomorphic mappings are introduced,such as strong and almost spirallike mappings of order α and type β,parabolic and spirallike mappings of type β and order ρ.It is easy to fi nd the examples of these new subclasses in C,which will enable us to better study the properties of these subclasses.While,it is very difficult in Cn.The introduction of the following Roper-Su ff ridge operator[1]

where z=(z1,z0)∈ Bn,z1∈ D,z0=(z2,···,zn)∈ Cn?1,f(z1)∈ H(D),a powerful way to construct biholomorphic mappings by means of univalent functions in C.Roper and Su ff ridge[1]proved the Roper-Su ff ridge operator preserves convexity and starlikeness on Bn.Graham and Kohr[2]proved the Roper-Su ff ridge operator preserves the properties of Bloch mappings on Bn.Therefore,we can construct a great deal of convex mappings or starlike mappings on Bnby corresponding functions on the unit disk D of C through the Roper-Su ff ridge operator.In recent years,the Roper-Su ff ridge extension operator was generalized on di ff erent domains and di ff erent spaces.The extended operators were proved to preserve some particular geometric properties such as starlikeness,convexity and spirallikeness(see[3–9]).

In 2016,Tang[10]introduced the generalized Roper-Su ff ridge operator

on Bergman-Hartogs domain which is based on the unit ball Bn,

where p>1,q>0,δ,l1,l2∈[0,1],γ∈,l1+δ≤1,l2+γ≤1,and proved the generalized Roper-Su ff ridge extension operators preserve the geometric properties of almost spirallike mapping of type β and order α,spirallike mappings of type β and order α and strongly spirallike mappings of type β.

Now,we extend the above operator to be

We mainly seek conditions under which the generalized Roper-Su ff ridge operator(1.1)preserves the properties of subclasses of biholomorphic mappings onIn Section 2,we give some de fi nitions and lemmas.In Section 3–6,we discuss the generalized operator(1.1)preserves the properties of(β,A,B),parabolic and spirallike mappings of type β and order ρ,strong and almost spirallike mappings of type β and order α as well as almost starlike mappings of complex order λ onunder di ff erent conditions,respectively.

which is discussed in[12–14].In the following,we will see an interesting phenomenon that(1.1)and(1.2)preserve the same geometric properties.

2 De fi nitions and Lemmas

In the following,let D denote the unit disk in C,Bndenote the unit ball in Cn.Let DF(z)denote the Fr′echet derivative of F at z.

To get the main results,we need the following de fi nitions and lemmas.

De fi nition 2.1(see[15]) Let ? be a bounded starlike circular domain in Cn.The Minkowski functional ρ(z)of ? is C1except for a lower-dimensional manifold.Suppose that F(z)is a normalized locally biholomorphic mapping on ?.Then we call F∈(β,A,B)with?1 ≤ A

Setting A= ?1= ?B?2α,A= ?B= ?α,B → 1?in De fi nition 2.1 respectively,we obtain the corresponding de fi nitions of spirallike mappings of type β and order α [13],strongly spirallike mappings of type β and order α [16],almost spirallike mappings of type β and order α [17]on ?.

3The Invariance of(β,A,B)

In addition,Lemma 2.8 tells us

if γj∈Hence|H(w,z)|?(?1+?2)<0,which follows F(w,z)∈(β,A,B).

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Remark 3.2Setting A=?B=?α and A=?1=?B?2α in Theorem 3.1 respectively,we obtain the corresponding results for strongly spirallike mappings of type β and order α,spirallike mappings of type β and order α.

4 The Invariance of Parabolic and Spirallike Mappings of Type β and Order ρ

which is proved to preserve the properties of parabolic and spirallike mappings of type β and order ρ in[18].So it is interesting that(4.1)and(4.9)preserve the same geometric properties.

Remark 4.2Setting ρ =0 and β =0 in Theorem 4.1 respectively,we get the corresponding results for parabolic and spirallike mappings of type β,parabolic and starlike mappings of order ρ.

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

Therefore,by Lemma 2.6,we get

From Lemma 2.8 we obtain

Remark 5.2Letting β =0 and α =0 in Theorem 5.1,respectively,we get the corresponding results for strongly spirallike mapping of type β and strong and almost starlike mappings of order α.If we only consider the last n components in(1.1),we get operator(1.2)which is proved to preserve the same geometric properties in[24–26].

6 The Invariance of Almost Starlike Mappings of Complex Order λ

Similar to Theorem 3.1,the left side of(6.1)is the real part of a holomorphic function so as to be a harmonic function.From the minimum principle of harmonic functions,we need only to prove that(6.1)holds for z ∈ ??.Thus ρ(w,z)=1.

Therefore,by(6.4)we obtain

which follows(6.1),thus F(w,z)is an almost starlike mapping of complex order λ on ?. ?

Remark 6.2If we only consider the last n components in(1.1),we can see that the properties of(1.1)discussed in Theorem 6.1 is the same to that of(1.2)discussed in[20].Letting λ =αα?1,α ∈ [0,1)in Theorem 6.1,we get the corresponding results for almost starlike mappings of order α.