# Geometric Mappings under the Perturbed Extension Operators in Complex Systems Analysis

Authors: Chaojun Wang; Yanyan Cui; Hao Liu
Journal: Discrete Dynamics in Nature and Society
 (2017)

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## Abstract

In this paper, we mainly seek conditions on which the geometric properties of subclasses of biholomorphic mappings remain unchanged under the perturbed Roper-Suffridge extension operators. Firstly we generalize the Roper-Suffridge operator on Bergman-Hartogs domains. Secondly, applying the analytical characteristics and growth results of subclasses of biholomorphic mappings, we conclude that the generalized Roper-Suffridge operators preserve the geometric properties of strong and almost spiral-like mappings of typeβ and order α, SΩ⁎(β,A,B) as well as almost spiral-like mappings of type β and order α under different conditions on Bergman-Hartogs domains. Sequentially we obtain the conclusions on the unit ball Bn and for some special cases. The conclusions include and promote some known results and provide new approaches to construct biholomorphic mappings which have special geometric characteristics in several complex variables.


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## Body

## 1. Introduction

The theory of several complex variables derives from the theory of one complex variable. There are many excellent results in geometric function theories of one complex variable. It is natural to think that we can extend these results in several complex variables, while some basic theorems (such as the models of the coefficients of the homogeneous expansion for biholomorphic functions being bounded on the unit disk [1]) are found not to hold in several complex variables. In 1933, Cartan [2] suggested that we can consider the geometric constraint of biholomorphic mappings, such as star-likeness and convexity. So many scholars devoted themselves to the research of star-like mappings and convex mappings. Recently many subclasses or expansions of star-like and convex mappings are introduced. The properties of biholomorphic mappings with special geometric properties are important research objects in geometric function theories of several complex variables. It is easy to find specific examples of these new subclasses or expansions in C, while it is very difficult in Cn. In order to study these subclasses better in several complex variables, we need the specific examples imminently.In 1995, Roper and Suffridge [3] introduced an operator (1)ϕnfz=fz1,f′z1z0′,where z=(z1,z0)∈Bn,z1∈D,z0=(z2,…,zn)∈Cn-1,f(z1)∈H(D),f′(0)=1. Roper and Suffridge proved the Roper-Suffridge operator preserves convexity and star-likeness on Bn. Graham et al. generalized the Roper-Suffridge operator and discussed the generalized operators preserving star-likeness and the block property in [4, 5]. In 2002, Graham et al. extended the Roper-Suffridge operator on the unit ball in Cn and proved the extended operator preserves star-likeness and convexity if and only if some conditions are satisfied in [6]. In 2003, Gong and Liu [7] generalized the Roper-Suffridge operator and obtained the generalized operator preserving ε star-likeness on the Reinhardt domain which leads to star-likeness and convexity in the cases of ε=0 and ε=1, respectively.All of the above illustrate that the Roper-Suffridge operator has good properties. Through the Roper-Suffridge operator or its generalizations we can construct lots of convex mappings and star-like mappings inCn by corresponding functions on the unit disk D of C. That will promote the development of the research of biholomorphic mappings. So the Roper-Suffridge operator plays an important role in several complex variables. In recent years, there are lots of results about the Roper-Suffridge extension operator which was generalized and modified on different domains in different spaces to preserve the geometric characteristics of convex mappings, star-like mappings, and their subclasses. Graham and Kohr gave a survey about the Roper-Suffridge extension operator and the developments in the theory of biholomorphic mappings in several complex variables to which it had led in [8].Muir and Suffridge [9] introduced the following generalized Roper-Suffridge extension operator: (2)Fz=fz1+f′z1Pz0,f′z1z0′,where f is a normalized biholomorphic function on the unit disk D,z=(z1,z0)′∈Bn,z1∈D,z0=(z2,…,zn)′∈Cn-1. The branch of the power function is chosen such that f′(0)=1. P:Cn-1→C is a homogeneous polynomial of degree 2. The extended operator (2) was proved to preserve star-likeness on P≤1/4 and convexity on P≤1/2 in [9] and was proved to take the extreme points of normalized convex functions to extreme points of normalized convex mappings of the Euclidean ball in Cn under precise conditions by Muir in [10]. Also the extended operator (2) was studied by Kohr and Muir in [11, 12] with Loewner chains. Later the operator (2) was generalized by Elin and Levenshtein and the generalized operator was proved to preserve the spiral-likeness property in [13] and was concluded that it can be embedded in a Loewner chain on the unit ball in Cn in [14]. Moreover, Elin and Levenshtein presented an extension operator for semigroup generators and concluded that the new one-dimensional covering results established in [13] are crucial. Furthermore, (2) was modified and discussed by Cui et al. in [15]. Elin introduced a general construction of the extension operators (3)ϕΓhx,y=hx,Γh,xy,where (x,y) is on the unit ball of the product X∗Y of two Banach spaces and Γ(h,x) is an operator-valued mapping which satisfies some natural conditions. ϕΓ[h](x,y) was proved to preserve star-likeness and spiral-likeness under some conditions in [16].Now, we introduce a new extension operator(4)Fw,z=w1fz1z1δ1,…,wsfz1z1δs,fz1+Gfz1z1γz0,fz1z1γz0′on the Bergman-Hartogs domain (5)Ωp1,…,ps,qBn=w1,…,ws,z∈Cm1×⋯×Cms×Bn:w12p1+⋯+ws2ps<KBnz,z-q,where (w,z)∈Ωp1,…,ps,qBn,z=(z1,z0)∈Bn, f(z1) is a normalized univalent holomorphic function on D, G is a holomorphic function in Cn-1 with G(0)=0,DG(0)=I,γ≥0, and the power functions take the branches such that [f(z1)/z1]δiz1=0=1(i=1,…,s) and [f(z1)/z1]γz1=0=1. The homogeneous expansion of G(z) is ∑j=0∞Pj(z), where Pj(z) is a homogeneous polynomial of degree j. Equation (4) is the modification of the extension operators discussed in [17].In the case ofw(1)=⋯=w(s)=0, (4) leads to the following operator: (6)Fz=fz1+Gfz1z1γz0,fz1z1γz0′,which can also be seen as the modification of the following generalized Roper-Suffridge extension operator introduced by Muir on the unit ball in complex Banach spaces, (7)ΦG,γfz=fz1+Gf′z1γz0,f′z1γz0′,where [f′(z1)]γz1=0=1. Muir proved [ΦG,γ(f)](z) is a Loewner chain preserving extension operator provided that G satisfies some conditions in [12].In this paper, we mainly discuss the invariance of several biholomorphic mappings under the generalized Roper-Suffridge extension operators (4) on the Bergman-Hartogs domains Ωp1,…,ps,qBn which is based on the unit ball Bn. In Section 2, we give some definitions and lemmas that are used to derive the main results. In Sections 3–5, we detailedly discuss the perturbed Extension Operator (4) preserving the geometric properties of strong and almost spiral-like mappings of type β and order α, SΩ∗(β,A,B), as well as almost spiral-like mappings of type β and order α under different conditions on Bergman-Hartogs domains and thus generalize the conclusions on the unit ball Bn in Cn. At last, we derive that the generalized Roper-Suffridge operators preserve the properties of subclasses of the three kinds of biholomorphic mappings mentioned above. The conclusions include and promote some known results.


## 2. Definitions and Lemmas

In the following, letD denote the unit disk in C and Bn denote the unit ball in Cn. Let DF(z) denote the Fréchet derivative of F at z.To get the main results, we need the following definitions and lemmas.Definition 1 (see [18]).
LetΩ be a bounded star-like circular domain in Cn. The Minkowski functional ρ(z) of Ω is C1 except for a lower-dimensional manifold. If f(z) is a normalized locally biholomorphic mapping on Ω, let α∈[0,1),β∈(-π/2,π/2), and (8)-α+itan⁡β1-α+1-itan⁡β1-α2ρz∂ρ∂zzDfz-1fz-1+c21-c2<2c1-c2.Then f(z) is called a strong and almost spiral-like mapping of type β and order α on Ω.Settingα=0, β=0, and α=β=0, Definition 1 reduces to the definition of strong spiral-like mappings of type β, strong and almost star-like mappings of order α, and strong star-like mappings, respectively.Definition 2 (see [19]).
LetΩ be a bounded star-like circular domain in Cn. The Minkowski functional ρ(z) of Ω is C1 except for a lower-dimensional manifold. Let F(z) be a normalized locally biholomorphic mapping on Ω. If (9)itan⁡β+1-itan⁡β2ρz∂ρ∂zzJF-1zFz-1-AB1-B2<B-A1-B2,z∈Ω∖0,where -1≤A<B<1 and β∈(-π/2,π/2), then we call F∈SΩ∗(β,A,B).SettingA=-1=-B-2α, A=-B=-α, and B→1- in Definition 2, respectively, we obtain the corresponding definitions of spiral-like mappings of type β and order α, strong spiral-like mappings of type β and order α, and almost spiral-like mappings of type β and order α on Ω.Definition 3 (see [20]).
LetΩ be a bounded star-like circular domain in Cn. The Minkowski functional ρ(z) of Ω is C1 except for a lower-dimensional manifold. Let f(z) be a normalized locally biholomorphic mapping on Ω. If (10)Re-iβ2ρz∂ρ∂zzDfz-1fz≥αcos⁡β,z∈Ω∖0,where α∈[0,1) and β∈(-π/2,π/2), then we call f(z) an almost spiral-like mapping of type β and order α.Settingα=0, β=0 in Definition 3, we obtain the definition of spiral-like mappings of type β and almost star-like mappings of order α on Bn, respectively.Lemma 4 (see [21]).
LetΩ be a bounded star-like circular domain in Cn. The Minkowski functional ρ(z) of Ω is C1 except for a lower-dimensional manifold Ω0. Then we have (11)2∂ρz∂zz=ρz,∂ρ∂zλz=∂ρz∂zλ≥0,∂ρ∂zeiθz=e-iθ∂ρz∂zθ∈R.Lemma 5 (see [12]).
LetP(z) be a homogeneous polynomial of degree m and let DP(z) be the Fréchet derivative of P at z. Then (12)DPzz=mPz.Lemma 6 (see [17]).
Letρ(w,z) be the Minkowski functional of Ωp1,…,ps,qBn. Let (w,z)∈∂Ωp1,…,ps,qBn. Then ρ(w,z)=1 and (13)∂ρw,z∂wij=piwi2pi-2wij¯2∇1+2∇2,∂ρw,z∂zi=∇1zi¯/z22∇1+2∇2,i=1,…,s,j=1,…,mi,where (14)∇1=n+1qπnqn!-q1-z2n+1q-1z2,∇2=∑k=1spkwk2pk.Lemma 7 (see [1]).
Iff(z) is a normalized biholomorphic function on the unit disk D, then (15)z1+z2≤fz≤z1-z2.Lemma 8 (see [22]).
Letf(z) be a strong spiral-like mapping of type α on bounded and balanced domain Ω with α∈(-π/2,π/2) and c∈(0,1). Let the Minkowski functional of Ω be ρ(z). Then (16)ρz1+cρz2≤ρfz≤ρz1-cρz2.Let f(z) be a strong spiral-like function of type α on D, then (17)z1+cz2≤fz≤z1-cz2.Lemma 9 (see [23]).
LetΩ⊂Cn be a bounded star-like circular domain and the Minkowski functional ρ(z) of Ω be C1 except for some submanifolds of lower dimensions. Let f(z)∈SΩ∗(A,B) be k-fold symmetric. Then(18)ρz1-AρzA-B/A≤ρfz≤ρz1+AρzA-B/A,A≠0ρze-Bρz≤ρfz≤ρzeBρz,A=0,or, equivalently,(19)z1-AzA-B/A≤fz≤z1+AzA-B/A,A≠0ze-Bz≤fz≤zeBz,A=0.The above estimates are all accurate.Lemma 10 (see [24]).
Letf(x) be an almost spiral-like mapping of type β and order α on the unit ball B in complex Banach spaces with α∈[0,1) and β∈(-π/2,π/2). Then (20)x1+1-2αx2α-1/2α-1≤fx≤x1-1-2αx2α-1/2α-1.For n=1 we get (21)z1+1-2αz2α-1/2α-1fz≤z1-1-2αz2α-1/2α-1.


## 3. The Invariance of Strong and Almost Spiral-Like Mappings of Typeβ and Order α

For simplicity, letΩ denote Ωp1,…,ps,qBn. In this section we will show that the perturbed Roper-Suffridge extension operator (4) preserves the geometric characteristics of strong and almost spiral-like mappings of type β and order α on Ω, and thus we obtain the conclusion on Bn; also we get the invariance of some subclasses.Theorem 11.
Letf(z1) be a strong and almost spiral-like function of type β and order α on D with α∈[0,1) and β∈(-π/2,π/2),c∈(0,1). Let F(w,z) be the mapping denoted by (4) with pi>1,δi∈[0,1](i=1,…,s),γ<1/4 and (f(z1)/z1)δiz1=0=1,(f(z1)/z1)γz1=0=1. Then F(w,z) is a strong and almost spiral-like mapping of type β and order α on Ω provided that Pj=0 for j<2/(1-4γ) and (22)q≥δn+1,∑j=2∞2j/2+1j-1Pj≤2c1-γ1-αcos⁡β1+c1+1-αcos⁡β,where δ=max{p1δ1,…,psδs}.Proof.
By Definition1, we need to prove(23)1-c22c1-itan⁡β1-α2ρw,z∂ρw,z∂w,zDFw,z-1Fw,z+1-c22c-α+itan⁡β1-α-1+c22c<1.It is obvious that (23) holds for w=z0=0. Otherwise, setting (w,z)=ζ(ξ,η)=|ζ|eiθ(ξ,η) where (ξ,η)∈∂Ω and ζ∈D¯∖{0}, by Lemma 4, we obtain (24)2ρw,z∂ρw,z∂w,zDFw,z-1Fw,z=2ρζeiθξ,η∂ρ∂w,zζeiθξ,ηDFζξ,ζη-1Fζξ,ζη=2ζe-iθ∂ρ∂w,zξ,ηDFζξ,ζη-1Fζξ,ζη=2∂ρ∂w,zξ,ηDFζξ,ζη-1Fζξ,ζηζ.Fix ξ,η, then 2∂ρ/∂w,zξ,η(DF(ζξ,ζη))-1F(ζξ,ζη)/ζ is holomorphic with respect to ζ. Due to the maximum modulus principle of holomorphic functions, the left side of (23) gets the maximum value at ζ=1. So we need only to prove that (23) holds for (w,z)∈∂Ω, which follows ρ(w,z)=1.
Let(25)qz1=1-c22c1-itan⁡β1-αfz1z1f′z1+1-c22c-α+itan⁡β1-α-1+c22c.Then qz1∈HD,qz1<1 and q(0)=-c. Letting g(z1)=2c/1-c2(q(z1)+c), we have g(z1)∈H(D),g(0)=0 and qz1=1-c2/2cgz1-c<1. In view of 1-c2/2cgz1-c<((1-c2)/2c)g(z1)-c, we obtain gz1<2c/(1-c)<1 provided that c≤1/3. By Schwarz Lemma we get gz1≤z1 which implies that (26)qz1+c≤1-c22cz1.
In addition, from (4) we have (27)DFw,z=fz1z1δ10⋯0v101×n-1⋯⋯⋯⋯⋯⋯00⋯fz1z1δsvs01×n-100⋯0u1∑j=2∞fz1z1γjDPjz00n-1×10n-1×1⋯0n-1×1γfz1z1γz1f′-fz1fz0′fz1z1γIn-1,where(28)vi=δiwifz1z1δiz1f′-fz1fi=1,…,s,u1=f′+z1f′-fz1f∑j=2∞γjfz1z1γjPjz0.Let (DF(w,z))-1F(w,z)=H(w,z)=h1,…,hs+n′, which follows DFw,zh1,…,hs+n′=F(w,z). Thus(29)fz1δihi+vihs+1=wifz1z1δi,i=1,…,su1hs+1+∑j=2∞fz1z1γjDPjz0hs+2,…,hs+n′=fz1+∑j=2∞fz1z1γjPjz0γfz1z1γz1f′-fz1fzjhs+1+fz1γhs+j=fz1z1γzj,j=2,…,n.Lemma 5 and a straightforward calculation show that(30)hi=wi1-δiz1f′-fz1f′1+1f∑j=2∞fz1z1γj1-jPjz0,i=1,…,shs+1=ff′+1f′∑j=2∞fz1z1γj1-jPjz0hs+j=zj1-γz1f′-fz1f′1+1f∑j=2∞fz1z1γj1-jPjz0,j=2,…,n.As a consequence of Lemma 6 we get (31)2∂ρ∂w,zw,zDFw,z-1Fw,z=Gw,z∇1+∇2,where (32)Gw,z=∑k=1spkwk2pk1-δkz1f′-fz1f′1+1f∑j=2∞fz1γj1-jPjz0+z12∇1z2fz1f′1+1f∑j=2∞fz1γj1-jPjz0+z02∇1z21-γz1f′-fz1f′1+1f∑j=2∞fz1γj1-jPjz0.
Let(33)1-c22c1-itan⁡β1-α2∂ρw,z∂w,zDFw,z-1Fw,z+1-c22c-α+itan⁡β1-α-1+c22c=I.Taking into account (25) and (31) we obtain the following equation string:(34)∇1+∇2I=1-c22c1-itan⁡β1-α∑k=1spkwk2pk1-δkz1f′-fz1f′1+1f∑j=2∞fz1γj1-jPjz0+z12∇1z2fz1f′1+1f∑j=2∞fz1γj1-jPjz0+z02∇1z21-γz1f′-fz1f′·1+1f∑j=2∞fz1γj1-jPjz0+1-c22c-α+itan⁡β1-α-1+c22c∇1+∇2=1-c22c1-itan⁡β1-α∇2+∇2~qz1+c1+1f∑j=2∞fz1γj1-jPjz0+z12∇1z2qz1-1-c22c-α+itan⁡β1-α+1+c22c1+1f∑j=2∞fz1γj1-jPjz0+z02∇1z21-c22c1-itan⁡β1-α+γqz1+c1+1f∑j=2∞fz1γj1-jPjz0+1-c22c-α+itan⁡β1-α-1+c22c∇1+∇2=-cz02∇1z2+∇2+∇2~qz1+c+z12z2∇1qz1+γz02z2∇1qz1+c1+1f∑j=2∞fz1γj1-jPjz0+z12z2∇11+c22c-1-c22c-α+itan⁡β1-α1f∑j=2∞fz1γj1-jPjz0=cγ-1z02∇1z2+c∇2~-∇2+∇2~+z12+γz02z2∇1qz1+∇2~+z12+γz02z2∇1qz1+c1f∑j=2∞fz1γj1-jPjz0+1-c22c1-itan⁡β1-αz12z2∇11f∑j=2∞fz1γj1-jPjz0,where ∇2~=∑k=1sδkpkwk2pk. We use (26) and hz1<1 to see that (35)∇1+∇2I-1<c1-γz02∇1z2+c∇2-∇2~+∇2~+z12+γz02z2∇1+1-c22c∇2~+z12+γz02z2∇1z1f∑j=2nfz1γjj-1Pjz0+1-c22c11-αcos⁡βz12z2∇11f∑j=2∞fz1γjj-1Pjz0-∇1+∇2≤c-11-γz02∇1z2+c-1∇2-∇2~+∇1z21-c22c∇2~∇1z2+z12+γz02+11-αcos⁡βz1f∑j=2∞fz1γjj-1Pjz0.
Note that(w,z)∈Ωp1,…,ps,qBn implies that (36)∑k=1swk2pk<KBnz,z-q=πnn!q1-z2n+1q,∇1=n+1qπnqn!-q1-z2n+1q-1z2,∇2~<δ∑k=1swk2pklead to(37)∇2~∇1<δ1-z2n+1qz2.Therefore(38)∇2~∇1z2+z12+γz02<δn+1q1-z2+z2+γ-1z02=δn+1q+1-δn+1qz2+γ-1z02<1provided that q≥δ/(n+1).
Lemma7 and (38) eventually lead us to the following inequality string: (39)∇1+∇2I-1<c-11-γz02∇1z2+∇1z21-c22c1+1-αcos⁡β1-αcos⁡βz1f∑j=2∞fz1γjj-1Pjz0≤c-11-γz02∇1z2+∇1z21-c22c1+1-αcos⁡β1-αcos⁡β1+z12∑j=2∞1-z1-2γjj-1Pjz0j=1-c22c1+1-αcos⁡β1-αcos⁡βz02∇1z2∑j=2∞1+z1j/2+11-z1j/2-2γj-1j-1Pj-2c1-γ1-αcos⁡β1+c1+1-αcos⁡β≤1-c22c1+1-αcos⁡β1-αcos⁡βz02∇1z2∑j=2∞2j/2+1j-1Pj-2c1-γ1-αcos⁡β1+c1+1-αcos⁡β≤0,where Pj=0 for j<2/(1-4γ) and (40)∑j=2∞2j/2+1j-1Pj≤2c1-γ1-αcos⁡β1+c1+1-αcos⁡β.This completes the proof.Settingw(1)=⋯=w(s)=0 in Theorem 11, the operator (4) reduces to the generalized Roper-Suffridge operator (6) and the Bergman-Hartogs domain Ω reduces to the unit ball Bn; thus, we draw the following conclusion on Bn in Cn.Corollary 12.
Letf(z1) be a strong and almost spiral-like function of type β and order α on D with α∈[0,1) and β∈(-π/2,π/2),c∈(0,1). Let F(z) be the mapping denoted by (6) with γ<1/4 and (f(z1)/z1)γz1=0=1. Then F(z) is a strong and almost spiral-like mapping of type β and order α on Bn provided that Pj=0 for j<2/(1-4γ) and (41)∑j=2∞2j/2+1j-1Pj≤2c1-γ1-αcos⁡β1+c1+1-αcos⁡β.Remark 13.
Settingα=0 and β=0 in Theorem 11 and Corollary 12, respectively, we get the corresponding results for strong and almost star-like mappings of order α and strong spiral-like mapping of type β.If we have the precise growth result of strong and almost spiral-like mapping of typeβ and order α and apply the growth result in Theorem 11, we can get more precise conclusion. But, up to now we only have the growth theorem of strong spiral-like function of type β [22]. In the following, we apply it in the process of discussing the invariance of strong spiral-like function of type β.Theorem 14.
Letf(z1) be a strong spiral-like function of type β on D with β∈(-π/2,π/2) and c∈(0,1). Let F(w,z) be the mapping denoted by (4) with pi>1,δi∈[0,1](i=1,…,s),γ<1/4 and (f(z1)/z1)δiz1=0=1,(f(z1)/z1)γz1=0=1. Then F(w,z) is a strong spiral-like mapping of type β and order α on Ω provided that Pj=0 for j<2/(1-4γ) and (42)q≥δn+1,∑j=2∞2j/2-1j-1Pj≤2c1-γcos⁡β1+c31+cos⁡β,where δ=max⁡{p1δ1,…,psδs}.Proof.
Settingα=0 in (39) and applying Lemma 8 we obtain the following inequality string: (43)∇1+∇2I-1<c-11-γz02∇1z2+∇1z21-c22c1+cos⁡βcos⁡βz1f∑j=2∞fz1γjj-1Pjz0≤c-11-γz02∇1z2+∇1z21-c22c1+cos⁡βcos⁡β1+cz12∑j=2∞1-cz1-2γjj-1Pjz0j≤1-c22c1+cos⁡βcos⁡βz02∇1z2∑j=2∞1+cz121+z1j/2-11-cz1j/2-2γj-1j-1Pj-2c1-γcos⁡β1+c1+cos⁡β≤1-c22c1+cos⁡βcos⁡βz02∇1z2∑j=2∞1+c22j/2-1j-1Pj-2c1-γcos⁡β1+c1+cos⁡β≤0,where Pj=0 for j<2/(1-4γ) and (44)∑j=2∞2j/2-1j-1Pj≤2c1-γcos⁡β1+c31+cos⁡β.Hence the assertion follows.


## 4. The Invariance ofSΩ∗(β,A,B)

As above, writeΩp1,…,ps,qBn=Ω. In this section, we mainly study the perturbed Extension Operator (4) preserving the geometric characteristics of SΩ∗(β,A,B) on Ω and then generalize the conclusion on Bn and get the invariance of some subclasses.Theorem 15.
Letf(z1)∈SD∗(β,A,B) with -1≤A<B<(A+1)/2<1 and β∈(-π/2,π/2). Let F(w,z) be the mapping denoted by (4) with pi>1,δi∈[0,1](i=1,…,s),γ∈(0,1/4) and (f(z1)/z1)δiz1=0=1,(f(z1)/z1)γz1=0=1. Then F(w,z)∈SΩ∗(β,A,B) provided that Pj=0 for j<2/(1-4γ) and (45)q≥δn+1,∑j=2∞2j/2+1j-1Pj≤1-γB-Acos⁡β1+B1+cos⁡β,where δ=max⁡{p1δ1,…,psδs}.Proof.
By Definition2, we need to prove (46)1-B2B-Aitan⁡β+1-itan⁡β2ρw,z∂ρw,z∂w,zDFw,z-1Fw,z-1-ABB-A<1.Similar to Theorem 11, we need only to prove that (46) holds for (w,z)∈∂Ω which implies ρ(w,z)=1.
Sincef(z1)∈SD∗(β,A,B), by Definition 2, we have (47)itan⁡β+1-itan⁡βfz1z1f′z1-1-AB1-B2<B-A1-B2.Let (48)1-B2B-Aitan⁡β+1-B2B-A1-itan⁡βfz1z1f′z1-1-ABB-A=hz1.Then h(z1)<1 and h(0)=-B. Letting g(z1)=(B-A)/(1-B2)(h(z1)+B), we get g(z1)∈H(D),g(0)=0 and |h(z1)|=(1-B2/B-A)gz1-B<1. In view of (1-B2/B-A)|g(z1)|-|B|≤|(1-B2/B-A)g(z1)-B|, we obtain |g(z1)|<(1+|B|)(B-A)/(1-B2)≤1 where -1≤A<B<(A+1)/2<1. By Schwarz Lemma, we get |g(z1)|≤|z1| which implies that (49)hz1+B≤1-B2B-Az1.
Let(50)1-B2B-Aitan⁡β+1-B2B-A1-itan⁡β2∂ρw,z∂w,zDFw,z-1Fw,z-1-ABB-A=I.Then, (31) and (48) lead to the following equation string: (51)∇1+∇2I=1-B2B-A1-itan⁡β∑k=1spkwk2pk1-δkz1f′-fz1f′1+1f∑j=2∞fz1γj1-jPjz0+z12∇1z2fz1f′1+1f∑j=2∞fz1γj1-jPjz0+1-B2B-Aitan⁡β-1-ABB-A∇1+∇2+z02∇1z21-γz1f′-fz1f′1+1f∑j=2∞fz1γj1-jPjz0=1-B2B-A1-itan⁡β∇2+∇2~hz1+B1+1f∑j=2∞fz1γj1-jPjz0+z12∇1z2hz1-1-B2B-Aitan⁡β+1-ABB-A1+1f∑j=2∞fz1γj1-jPjz0+z02∇1z21-B2B-A1-itan⁡β+γhz1+B1+1f∑j=2∞fz1γj1-jPjz0+1-B2B-Aitan⁡β-1-ABB-A∇1+∇2=Bγ-1z02∇1z2+B∇2~-∇2+∇2~+∇1z2z12+γz02hz1+∇2~hz1+B+z12∇1z2hz1-1-B2B-Aitan⁡β+1-ABB-A+γz02∇1z2hz1+B1f∑j=2∞fz1γj1-jPjz0=Bγ-1z02z2∇1+B∇2~-∇2+∇2~+∇1z2z12+γz02hz1+∇2~+∇1z2z12+γz02hz1+B1f∑j=2∞fz1γj1-jPjz0+1-B2B-A1-itan⁡βz12∇1z21f∑j=2∞fz1γj1-jPjz0.By (49), (38), and Lemma 7 there comes the following inequality string: (52)∇1+∇2I-1<B1-γz02z2∇1+B∇2-∇2~+∇2~+∇1z2z12+γz02-∇1+∇2+1-B2B-A∇2~+∇1z2z12+γz02z1f∑j=2∞fz1γjj-1Pjz0+1-B2B-Acos⁡βz1∇1z21f∑j=2∞fz1γjj-1Pjz0=B-11-γz02z2∇1+B-1∇2-∇2~+1-B2B-A∇1z2∇2~∇1z2+z12+γz02+1cos⁡βz1f∑j=2∞fz1γjj-1Pjz0<B-11-γz02z2∇1+1-B2B-A1+cos⁡βcos⁡β∇1z2z1f∑j=2∞fz1γjj-1Pjz0j=1-B2B-A1+cos⁡βcos⁡βz02∇1z2z1f∑j=2∞fz1γjj-1Pjz0j-2-1-γB-Acos⁡β1+B1+cos⁡β≤1-B2B-A1+cos⁡βcos⁡βz02∇1z2∑j=2∞1+z1j/2+11-z1j/2-2γj-1j-1Pj-1-γB-Acos⁡β1+B1+cos⁡β≤1-B2B-A1+cos⁡βcos⁡βz02∇1z2∑j=2∞2j/2+1j-1Pj-1-γB-Acos⁡β1+B1+cos⁡β≤0,where Pj=0 for j<2/(1-4γ) and (53)∑j=2∞2j/2+1j-1Pj≤1-γB-Acos⁡β1+B1+cos⁡β.Therefore (46) holds, which follows F(w,z)∈SΩ∗(β,A,B).Settingw(1)=⋯=w(s)=0 in Theorem 15, we draw the following invariance of the mappings SBn∗(β,A,B) under the generalized Roper-Suffridge operator (6) on the unit ball Bn in Cn.Corollary 16.
Letf(z1)∈SD∗(β,A,B) with -1≤A<B<(A+1)/2<1 and β∈(-π/2,π/2). Let F(z) be the mapping denoted by (6) with γ∈(0,1/4) and (f(z1)/z1)γz1=0=1. Then F(z)∈SBn∗(β,A,B) provided that Pj=0 for j<2/(1-4γ) and (54)∑j=2∞2j/2+1j-1Pj≤1-γB-Acos⁡β1+B1+cos⁡β.In the process of discussing the invariance ofSΩ∗(A,B), if we apply the growth theorem of SΩ∗(A,B) in [23], we can get the following more precise conclusion.Theorem 17.
Letf(z1)∈SD∗(A,B) with -1≤A<B<(A+1)/2<1. Let F(w,z) be the mapping denoted by (4) with pi>1,δi∈[0,1](i=1,…,s),γ∈(0,1/4) and (f(z1)/z1)δiz1=0=1,(f(z1)/z1)γz1=0=1. Let q≥δ/(n+1) where δ=max⁡{p1δ1,…,psδs}. Then we have the following conclusions:(1)
ForA≠0, F(w,z)∈SΩ∗(A,B) provided that (55)∑j=2∞1-AA-B/A1+AB-A/Aγjj-1Pj≤1-γB-A21+B.(2)
ForA=0 and B>0, F(w,z)∈SΩ∗(A,B) provided that (56)∑j=2∞eB1+γjj-1Pj≤1-γB-A21+B.(3)
ForA=0 and B<0, F(w,z)∈SΩ∗(A,B) provided that (57)∑j=2∞j-1Pj≤1-γB-A21+B.Proof.
Settingβ=0 in the proof of Theorem 15, by using (52) and Lemma 9 we obtain the following conclusions:(1)
For the case thatA≠0 we get (58)∇1+∇2I-1<1-B2B-A1+cos⁡βcos⁡βz02∇1z2z1f∑j=2∞fz1γjj-1Pjz0j-2-1-γB-Acos⁡β1+B1+cos⁡β≤1-B2B-A1+cos⁡βcos⁡βz02∇1z2∑j=2∞1-Az1A-B/A1+Az1B-A/Aγj1-z12j/2-1j-1Pj-1-γB-A21+B≤1-B2B-A1+cos⁡βcos⁡βz02∇1z2∑j=2∞1-AA-B/A1+AB-A/Aγjj-1Pj-1-γB-A21+B≤0,where (59)∑j=2∞1-AA-B/A1+AB-A/Aγjj-1Pj≤1-γB-A21+B.(2)
For the case thatA=0 and B>0 we have (60)∇1+∇2I-1<1-B2B-A1+cos⁡βcos⁡βz02∇1z2z1f∑j=2∞fz1γjj-1Pjz0j-2-1-γB-Acos⁡β1+B1+cos⁡β≤1-B2B-A1+cos⁡βcos⁡βz02∇1z2∑j=2∞eBz11+γj1-z12j/2-1j-1Pj-1-γB-A21+B≤1-B2B-A1+cos⁡βcos⁡βz02∇1z2∑j=2∞eB1+γjj-1Pj-1-γB-A21+B≤0,where (61)∑j=2∞eB1+γjj-1Pj≤1-γB-A21+B.(3)
For the case thatA=0 and B<0, similar to (2), we get (62)∇1+∇2I-1<1-B2B-A1+cos⁡βcos⁡βz02∇1z2∑j=2∞eBz11+γj1-z12j/2-1j-1Pj-1-γB-A21+B≤1-B2B-A1+cos⁡βcos⁡βz02∇1z2∑j=2∞j-1Pj-1-γB-A21+B≤0,where (63)∑j=2∞j-1Pj≤1-γB-A21+B.From (1) to (3) we get the desired conclusion.Remark 18.
SettingA=-1=-B-2α and A=-B=-α in Theorems 15 and 17 and Corollary 16, respectively, we get the corresponding results for spiral-like mappings of type β and order α and strong spiral-like mappings of type β and order α.


## 5. The Invariance of Almost Spiral-Like Mappings of Typeβ and Order α

In the following, we mainly discuss the perturbed Extension Operator (4) preserving the geometric characteristics of almost spiral-like mappings of type β and order α on Ω, and thus we get the conclusion on Bn as well as the results about some subclasses.Theorem 19.
Letf(z1) be an almost spiral-like function of type β and order α on D with α∈[0,1/2)∪(1/2,1) and β∈(-π/2,π/2). Let F(w,z) be the mapping denoted by (4) with pi>1,δi∈[0,1](i=1,…,s),γ≥0 and (f(z1)/z1)δiz1=0=1,(f(z1)/z1)γz1=0=1. Let Pj=0(j=2,3) and q≥δ/(n+1) where δ=max⁡{p1δ1,…,psδs}. Then F(w,z) is an almost spiral-like mapping of type β and order α on Ω provided that (64)∑j=4∞j-1Pj2-2α2α-1/2α-12α21-α/2α-1γj2j/2-1≤1-α1-γcos⁡β21-αcos⁡β+1.Proof.
By Definition3, we need to prove (65)R2ρw,z∂ρw,z∂w,ze-iβDFw,z-1Fw,z-αcos⁡β≥0,which is equivalent to (66)R1-itan⁡β2ρw,z∂ρw,z∂w,zDFw,z-1Fw,z-α≥0.Similar to Theorem 11, the left side of (66) is the real part of a holomorphic mapping and thus is a harmonic function. Due to the minimum principle of harmonic functions, we need only to prove that (66) holds for z∈∂Ω which implies that ρ(w,z)=1.
Let(67)1-itan⁡βfz1f′-α=hz1.Since f(z1) is an almost spiral-like function of type β and order α on D, then h(z1)∈H(D),Rh(z1)>0,h(0)=1-itan⁡β-α. Therefore (h(z1)-(1-α-itan⁡β))/(h(z1)+(1-α+itan⁡β))<1. Let (68)gz1=hz1-1-α-itan⁡βhz1+1-α+itan⁡β.Then |g(z1)|<1,g(0)=0. Moreover Rh(z1)>0 implies h(z1)≠-(1-α+itan⁡β); thus g(z1)∈H(D). Applying Schwarz Lemma we obtain |g(z1)|≤|z1|; in other words, (69)1+21-αhz1-1-α-itan⁡β≥1z1,which leads to (70)hz1-1-α-itan⁡β≤21-αz11-z1.
Let(71)1-itan⁡β2∂ρw,z∂w,zDFw,z-1Fw,z-α=I.We then get the following equation string by (31) and (67): (72)∇1+∇21-itan⁡β2∂ρw,z∂w,zDFw,z-1Fw,z-α=1-itan⁡β∑k=1spkwk2pk1-δkz1f′-fz1f′1+1f∑j=2∞fz1γj1-jPjz0+z12∇1z2fz1f′1+1f∑j=2∞fz1γj1-jPjz0+z02∇1z21-γz1f′-fz1f′1+1f∑j=2∞fz1γj1-jPjz0-α∇1+∇2=1-itan⁡β∇2+∇2~hz1-1+itan⁡β+α1+1f∑j=2∞fz1γj1-jPjz0+z12∇1z2hz1+α1+1f∑j=2∞fz1γj1-jPjz0-α∇1+∇2+z02∇1z21-itan⁡β+γhz1-1+itan⁡β+α1+1f∑j=2∞fz1γj1-jPjz0=1-α-itan⁡β1-γz02∇1z2+1-α-itan⁡β∇2-∇2~+∇2~+∇1z2z12+γz02hz1+∇2~hz1-1+α+itan⁡β+z12∇1z2hz1+α+γz02z2∇1hz1-1+α+itan⁡β1f∑j=2∞fz1γj1-jPjz0=1-α-itan⁡β1-γz02∇1z2+1-α-itan⁡β∇2-∇2~+∇2~+∇1z2z12+γz02hz1+∇2~+∇1z2z12+γz02hz1-1+α+itan⁡β·1f∑j=2∞fz1γj1-jPjz0+1-itan⁡β∇1z12z21f∑j=2∞fz1γj1-jPjz0.Therefore, by (70), (38), and Lemma 10 there comes the following inequality string: (73)∇1+∇2R1-itan⁡β2∂ρw,z∂w,zDFw,z-1Fw,z-α=1-α1-γz02∇1z2+1-α∇2-∇2~+∇2~+∇1z2z12+γz02Rhz1+R∇2~+∇1z2z12+γz02hz1-1+α+itan⁡β1f∑j=2∞fz1γj1-jPjz0+R1-itan⁡β∇1z12z21f∑j=2∞fz1γj1-jPjz0≥1-α1-γz02∇1z2-∇2~+∇1z2z12+γz0221-α1-z1+1cos⁡β∇1z2·z1f∑j=2∞fz1γjj-1Pjz0j=1-α1-γz02∇1z2-∇1z211-z1∇2~∇1z2+z12+γz0221-α+1-z1cos⁡β·z1f∑j=2∞fz1γjj-1Pjz0j≥1-α1-γz02∇1z2-∇1z211-z121-αcos⁡β+1cos⁡βz1f∑j=2∞fz1γjj-1Pjz0j≥1-α1-γz02∇1z2-∇1z211-z121-αcos⁡β+1cos⁡βz1f∑j=2∞fz1γjj-1Pjz0j≥z02∇1z21-α1-γ-21-αcos⁡β+1cos⁡β∑j=2∞j-1Pj·1+1-2αz12α-1/2α-11-1-2αz121-α/2α-1γj1+z1j/2-11-z1j/2-2≥z02∇1z21-α1-γ-21-αcos⁡β+1cos⁡β∑j=2∞j-1Pj·2-2α2α-1/2α-12α21-α/2α-1γj2j/2-1,where Pj=0(j=2,3) and (74)∑j=4∞j-1Pj2-2α2α-1/2α-12α21-α/2α-1γj2j/2-1≤1-α1-γcos⁡β21-αcos⁡β+1.This is equal to (66) and hence completes the proof.Settingw(1)=⋯=w(s)=0 in Theorem 19, we draw the following invariance of almost spiral-like mappings of type β and order α under the generalized Roper-Suffridge operator (6) on the unit ball Bn in Cn.Corollary 20.
Letf(z1) be an almost spiral-like function of type β and order α on D with α∈[0,1/2)∪(1/2,1) and β∈(-π/2,π/2). Let F(z) be the mapping denoted by (6) with γ≥0 and (f(z1)/z1)γz1=0=1. Let Pj=0 for j=2,3. Then F(z) is an almost spiral-like mapping of type β and order α on Bn provided that (75)∑j=4∞j-1Pj2-2α2α-1/2α-12α21-α/2α-1γj2j/2-1≤1-α1-γcos⁡β21-αcos⁡β+1.Remark 21.
Settingα=0 and β=0 in Theorem 19 and Corollary 20, respectively, we get the corresponding results for spiral-like mappings of type β and almost star-like mappings of order α.



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