In this article, the generalized Roper-Suffridge extension operator in Banach spaces for locally biholomorphic mappings is introduced. It is proved that this operator preserves the starlikeness on some domains in Bana...In this article, the generalized Roper-Suffridge extension operator in Banach spaces for locally biholomorphic mappings is introduced. It is proved that this operator preserves the starlikeness on some domains in Banach spaces but does not preserves convexity for some cases. Moreover, the growth theorem, covering theorem, and the radius of starlikeness are discussed. Some results of Roper and Suffridge, Gong and Liu, Graham et al in C^n are extended to Hilbert spaces or Banach spaces.展开更多
In this paper, we introduce the generalized R oper-Suffridge extension operator for locally biholomorphic mappings. It is sh own that this operator preserves the starlikeness on some Reinhardt domains and does not pre...In this paper, we introduce the generalized R oper-Suffridge extension operator for locally biholomorphic mappings. It is sh own that this operator preserves the starlikeness on some Reinhardt domains and does not preserve convexity for some cases. Meanwhile, the growth theorem and di stortion theorem of the corresponding mappings are given.展开更多
In this article,the authors obtain an inequality of homogeneous expansion for f,where f is a quasi-convex mapping(including quasi-convex mapping of type A and quasi-convex mapping of type B)defined on the open unit po...In this article,the authors obtain an inequality of homogeneous expansion for f,where f is a quasi-convex mapping(including quasi-convex mapping of type A and quasi-convex mapping of type B)defined on the open unit polydisk in C n .Meanwhile,the authors also investigate its application.展开更多
In this article, a normalized biholomorphic mapping f defined on bounded starlike circular domain in Cn is considered, where z = 0 is a zero of order k ＋ 1 of f（z） - z. The sharp growth, covering theorems for almos...In this article, a normalized biholomorphic mapping f defined on bounded starlike circular domain in Cn is considered, where z = 0 is a zero of order k ＋ 1 of f（z） - z. The sharp growth, covering theorems for almost starlike mappings of order α and starlike mappings of order α are established. Meanwhile, the construction of the above mappings on bounded starlike circular domain in Cn is also discussed, it provides the extremal mappings for the growth, covering theorems of the above mappings.展开更多
In this paper,we first establish the sharp growth theorem and the distortion theorem of the Frechet derivative for biholomorphic mappings defined on the unit ball of complex Banach spaces and the unit polydisk in C^(n...In this paper,we first establish the sharp growth theorem and the distortion theorem of the Frechet derivative for biholomorphic mappings defined on the unit ball of complex Banach spaces and the unit polydisk in C^(n) with some restricted conditions.We next give the distortion theorem of the Jacobi determinant for biholomorphic mappings defined on the unit ball of C^(n) with an arbitrary norm and the unit polydisk in C^(n) under certain restricted assumptions.Finally we obtain the sharp Goluzin type distortion theorem for biholomorphic mappings defined on the unit ball of complex Banach spaces and the unit polydisk in C^(n) with some additional conditions.The results derived all reduce to the corresponding classical results in one complex variable,and include some known results from the prior literature.展开更多
We mainly discuss the invariance of some subclasses of biholomorphic mappings under the generalized Roper-Suffridge operators on Bergman-Hartogs domains which are based on the unit ball Bn. Using the geometric propert...We mainly discuss the invariance of some subclasses of biholomorphic mappings under the generalized Roper-Suffridge operators on Bergman-Hartogs domains which are based on the unit ball Bn. Using the geometric properties and the distortion results of subclasses of biholomorphic mappings, we obtain the geometric characters of almost spirallike mappings of type β and order α, S_?~*(β, A, B), strong and almost spirallike mappings of type βand order α maintained under the generalized Roper-Suffridge operators on Bergman-Hartogs domains. Sequentially, we conclude that the generalized operators and the known operators preserve the same properties under some conditions. The conclusions generalize some known results.展开更多
The construction of normalized biholomorphic convex mappings in the Reinhardt domain $D_p = \{ (z_1 ,z_2 , \cdots ,z_n ) \in \mathbb{C}^n :\left| {z_1 } \right|^{p_1 } + \left| {z_2 } \right|^{p_2 } + \cdots + \left| ...The construction of normalized biholomorphic convex mappings in the Reinhardt domain $D_p = \{ (z_1 ,z_2 , \cdots ,z_n ) \in \mathbb{C}^n :\left| {z_1 } \right|^{p_1 } + \left| {z_2 } \right|^{p_2 } + \cdots + \left| {z_n } \right|^{p_n } < 1\} $ , p j > 2, j = 1,2,?, n) of ? n is discussed. The authors set up a Decomposition Theorem for such mappings. As a special case, it is proved that, for each such mapping f, the first k-terms of the homogeneous expansion of its j-th component f j , j = 1, 2, ?, n, depends only on z j , where k is the number that satisfies k < min {p 1, p 2,?, p n ≤ k + 1. When p1,p2, ... ,pn → ∞ , this derives the Decomposition Theorem of normalized biholomorphic convex mappings in the polydisc which was gotten by T.J. Suffridge in 1970.展开更多
The power series expansions of normalized biholomorphic convex mappings on the Reinhardt domain are studied.It is proved that the first (k+1 ) terms ofthe expansions of the jth component fj of such a map / depend only...The power series expansions of normalized biholomorphic convex mappings on the Reinhardt domain are studied.It is proved that the first (k+1 ) terms ofthe expansions of the jth component fj of such a map / depend only on zj for 1≤j≤n,where k is the natural number that satisfies k<p≤k+1.When p∞,this gives the result on the unit polydisc obtained by Sulfridge in 1970.展开更多
The estimations of the bounds of the Bloch constant of locally biholomorphic mappingson irreducible bounded symmetric domains are given. When the domain is a unit circle, theestimation of the lower bounds is just the ...The estimations of the bounds of the Bloch constant of locally biholomorphic mappingson irreducible bounded symmetric domains are given. When the domain is a unit circle, theestimation of the lower bounds is just the famous one-half estimation.展开更多
In this paper, we generalize the Roper–Suffridge operator on the extended Hartogs domains.By using the geometric properties and the growth theorems of subclasses of biholomorphic mappings,we obtain the generalized op...In this paper, we generalize the Roper–Suffridge operator on the extended Hartogs domains.By using the geometric properties and the growth theorems of subclasses of biholomorphic mappings,we obtain the generalized operators preserve the properties of parabolic and spirallike mappings of type β and order ρ, S*_Ω(β, A, B), almost starlike mapping of complex order λ on ΩN under different conditions, and thus we get the corresponding results on the unit ball B^n in C^n. The conclusions lead to some known results.展开更多
Let Gi be a closed Lie subgroup of U（n）, Ωi be a bounded Gi-invariant domain in Cn which contains 0, and （9（Cn）Gi = C, for i= 1,2. If f ： f21 →2 is abiholomorphism, and f（0） = 0, then f is a polynomial mappi...Let Gi be a closed Lie subgroup of U（n）, Ωi be a bounded Gi-invariant domain in Cn which contains 0, and （9（Cn）Gi = C, for i= 1,2. If f ： f21 →2 is abiholomorphism, and f（0） = 0, then f is a polynomial mapping （see Ning et al. （2017））. In this paper, we provide an upper bound for the degree of such polynomial mappings. It is a natural generalization of the well-known Cartan＇s theorem.展开更多
The distortion theorem for biholomorphic staxlike mappings(with respect to origin) inbounded symmetric domains are given.The distortion theorem for locally biholomorphic convexmappings in bounded symmetric domains are...The distortion theorem for biholomorphic staxlike mappings(with respect to origin) inbounded symmetric domains are given.The distortion theorem for locally biholomorphic convexmappings in bounded symmetric domains are given also.展开更多
1. In [1], T. J. Suffridge gave the necessary and sufficient condition for local biholomorphic mappings on the Reinhardt domains , to be starlike. Using this condition Yu Qihuang et al. gave the growth theorem and cov...1. In [1], T. J. Suffridge gave the necessary and sufficient condition for local biholomorphic mappings on the Reinhardt domains , to be starlike. Using this condition Yu Qihuang et al. gave the growth theorem and covering theorem for normalized biholomorphic starlike mappings in D_p. In this note we give a necessary and suffident condition for local biholomorphic mappings on more general Reinhardt展开更多
The paper is made of two parts.In first part,We give the growth and 1/4-theorems for spiral like maps on the unit ball in l^p.Particularly,corresponding results were given in B^p.In the second part,we give the growth ...The paper is made of two parts.In first part,We give the growth and 1/4-theorems for spiral like maps on the unit ball in l^p.Particularly,corresponding results were given in B^p.In the second part,we give the growth and 1/4-theorems for spirallike maps in an inner product space.We prove that the results is best.展开更多
In this note, the author introduces some new subcIasses of starlike mappings S^＊Ωn1p2,…,pn（β,A,B）={f∈H（Ω）：｜itanβ＋（1-itanβ）2/p（z）аp/аz（z）Jf^-1（z）f（z）-1-AB/1-B^2｜〈B-A/1-B^2},on Reinhardt dom...In this note, the author introduces some new subcIasses of starlike mappings S^＊Ωn1p2,…,pn（β,A,B）={f∈H（Ω）：｜itanβ＋（1-itanβ）2/p（z）аp/аz（z）Jf^-1（z）f（z）-1-AB/1-B^2｜〈B-A/1-B^2},on Reinhardt domains Ωn1p2,…,pn=z∈C^n：｜z1｜^2＋n∑j=2｜zj｜^pj〈1}where - 1≤A〈B〈1,q=min{p2,…,pn}≥1,l=max{p2,…,pn}≥2 and β ∈（-π/2,π/2）.Some different conditions for P are established such that these classes are preserved under the following modified Roper-Suffridge operator F（z）=（f（z1）＋f＇（z1）Pm（z0）,（f＇（z1））^1/mz0）＇where f is a normalized biholomorphic function on the unit disc D, z = （z1,z0） ∈Ωn1p2,…,pn,z0=（z2,…,zn）∈ C^n-1.Another condition for P is also obtained such that the above generalized Roper-Suffridge operator preserves an almost spirallike function of type/3 and order β These results generalize the modified Roper-Suffridge extension oper-ator from the unit ball to Reinhardt domains. Notice that when p2 = p3 …=pn = 2,our results reduce to the recent results of Feng and Yu.展开更多
In this article, first, a sufficient condition for a starlike mapping of order a f（x） defined on the unit ball in a complex Banach space is given. Second, the sharp estimate of the third homogeneous expansion for f ...In this article, first, a sufficient condition for a starlike mapping of order a f（x） defined on the unit ball in a complex Banach space is given. Second, the sharp estimate of the third homogeneous expansion for f is established as well, where f（z） = （f1（z）, f2（z）,..., fn（z））＇ is a starlike mapping of order a or a normalized biholomorphic starlike mapping defined on the unit polydisk in Cn, and D2fk（0）（z2） /2i= zk（∑l=1^b akzzl）, k = 2t l=1 k = 1, 2,..., n. Our result states that the Bieberbaeh conjecture in several complex variables （the case of the third homogeneous expansion for starlike mappings of order α and biholomorphic starlike mappings） is partly proved.展开更多
This article is devoted to a deep study of the Roper-Suffridge extension operator with special geometric properties.First,we prove that the Roper-Suffridge extension operator preservesϵstarlikeness on the open unit ba...This article is devoted to a deep study of the Roper-Suffridge extension operator with special geometric properties.First,we prove that the Roper-Suffridge extension operator preservesϵstarlikeness on the open unit ball of a complex Banach space C×X,where X is a complex Banach space.This result includes many known results.Secondly,by introducing a new class of almost boundary starlike mappings of orderαon the unit ball B n of C n,we prove that the Roper-Suffridge extension operator preserves almost boundary starlikeness of orderαon B n.Finally,we propose some problems.展开更多
In this paper, we construct a new Roper-Suffridge extension operator Φn^r,β1,,βn（f）（z） = F（z） = （（rf（z1/r）/z1）^β1z1,（rf（z1/r）/z1）^β2z2,...,（rf（z1/r）/z1）^βnzn）＇,where f is a normalized locall...In this paper, we construct a new Roper-Suffridge extension operator Φn^r,β1,,βn（f）（z） = F（z） = （（rf（z1/r）/z1）^β1z1,（rf（z1/r）/z1）^β2z2,...,（rf（z1/r）/z1）^βnzn）＇,where f is a normalized locally biholomorphic function on the unit disc D, r = sup{｜z1｜ ： z =（z1, ···, zn） ∈Ω}, β1∈ [0, 1], 0 ≤βk≤β1, k = 2, ···, n, then we prove it can preserve the property of spirallikeness of type β, almost starlikeness of order α and starlikeness of orderα on bounded complete Reinhardt domain Ω, respectively.展开更多
This note is devoted to applying the principle of subordination in order to explore the Roper-Suffridge extension operator and the Pfaltzgraff-Suffridge extension operator with special analytic properties.First,we pro...This note is devoted to applying the principle of subordination in order to explore the Roper-Suffridge extension operator and the Pfaltzgraff-Suffridge extension operator with special analytic properties.First,we prove that both the Roper-Suffridge extension operator and the Pfaltzgraff-Suffridge extension operator preserve subordination.As applications,we obtain that if β∈[0,1],γ∈[0,1/r]and β+γ≤1,then the Roper-Suffridge extension operator Φ_(β+γ)(f)(z)=(f(z_(1)),(f(z_(1))/z_(1))^(β)(f’(z_(1)))^(γ)w),z∈Ω_(p,r) preserves an almost starlike mapping of complex order λ on Ω_(p,r)={z=(z_(1),w)∈C×X:|z_(1)|^(p)+‖w‖_(X)^(r)<1},where 1≤p≤2,r≥1 and X is a complex Banach space.Second,by applying the principle of subordination,we will prove that the Pfaltzgraff-Suffridge extension operator preserves an almost starlike mapping of complex order λ.Finally,we will obtain the lower bound of distortion theorems associated with the Roper-Suffridge extension operator.This subordination principle seems to be a new idea for dealing with the Loewner chain associated with the Roper-Suffridge extension operator,and enables us to generalize many known results from p=2 to 1≤p≤2.展开更多
The generalized Roper-Suffridge extension operatorΦ(f) on the bounded complete Rein- hardt domainΩin C^n with n≥2 is defined byΦ_(n,β_2,γ_2,...,β_n,γ_n)~r(f)(z)=rf((z_1)/r),((rf(z_1)/r)/(z_1))^(β2)(f′((z_1)/...The generalized Roper-Suffridge extension operatorΦ(f) on the bounded complete Rein- hardt domainΩin C^n with n≥2 is defined byΦ_(n,β_2,γ_2,...,β_n,γ_n)~r(f)(z)=rf((z_1)/r),((rf(z_1)/r)/(z_1))^(β2)(f′((z_1)/r)^(γ2)z_2,...,((rf(z_1)/r)/(z_1))^(β_n)(f′((z_1)/r))^(γ_n)z_n) for (z_1,z_2,...,z_n)∈Ω,where r=r(Ω) = sup{|z_1|:(z_1,z_2,...,z_n)∈Ω},0≤γ_j≤1-β_j,0≤β_j≤1, and we choose the branch of the power functions such that ((f(z_1))/(z_1))^(β_j)|_(z_1=0)=1 and (f′(z_1))^(γ_j)|_(z_1=0)= 1,j=2,...,n.In this paper,we prove that the operatorΦ_(n,β_2,γ_2,...,β_n,γ_n)~r(f) is from the subset of S_α~*(U) to Sα~*(Ω)(0≤α<1) onΩand the operatorΦ_(n,β_2,γ_2,...,β_n,γ_n)~r(f) preserves the starlikeness of order a or the spirallikeness of typeβon D_p for some suitable constantsβ_j,γ_j,p_j,where D_p= {(z_1,z_2,...,z_n)∈C^n:∑_(j=1)~n|z_j|^(p_j)<1}(p_j>0,j=1,2...,n),U is the unit disc in the complex plane C,and S_α~* (Ω) is the class of all normalized starlike mappings of orderαonΩ.We also obtain thatΦ_(n,β_2,γ_2,...,β_n,γ_n)~r(f)∈S_α~*(D_p) if and only if f∈S_α~*(U) for 0≤α<1 and some suitable constantsβ_j,γ_j,p_j.展开更多
基金This research is partly supported by the National Natural Science Foundation of China (10471048) the Doctoral Foundation of the Education Committee of China(20050574002)+1 种基金 the Natural Science Foundation of Fujian Province, China (Z0511013)the Education Commission Foundation of Fujian Province, China (JB04038)
文摘In this article, the generalized Roper-Suffridge extension operator in Banach spaces for locally biholomorphic mappings is introduced. It is proved that this operator preserves the starlikeness on some domains in Banach spaces but does not preserves convexity for some cases. Moreover, the growth theorem, covering theorem, and the radius of starlikeness are discussed. Some results of Roper and Suffridge, Gong and Liu, Graham et al in C^n are extended to Hilbert spaces or Banach spaces.
文摘In this paper, we introduce the generalized R oper-Suffridge extension operator for locally biholomorphic mappings. It is sh own that this operator preserves the starlikeness on some Reinhardt domains and does not preserve convexity for some cases. Meanwhile, the growth theorem and di stortion theorem of the corresponding mappings are given.
基金Sponsored by National Natural Science Foundation of China under grant No.10571164Specialized Research Fund for the Doctoral Program of Higher Education under grant No.20050358052Guangdong Natural Science Foundation under grant No.06301315
文摘In this article,the authors obtain an inequality of homogeneous expansion for f,where f is a quasi-convex mapping(including quasi-convex mapping of type A and quasi-convex mapping of type B)defined on the open unit polydisk in C n .Meanwhile,the authors also investigate its application.
基金The research was supported by the National Nat ural Science Foundation of China(10571164)Specialized Research Fund for the Doctoral Program of Higher Education(20050358052)+1 种基金Guangdong Natural Science Foundation(06301315)the Doctoral Foundation of Zhanjiang Normal University(Z0420)
文摘In this article, a normalized biholomorphic mapping f defined on bounded starlike circular domain in Cn is considered, where z = 0 is a zero of order k ＋ 1 of f（z） - z. The sharp growth, covering theorems for almost starlike mappings of order α and starlike mappings of order α are established. Meanwhile, the construction of the above mappings on bounded starlike circular domain in Cn is also discussed, it provides the extremal mappings for the growth, covering theorems of the above mappings.
基金Supported by National Natural Science Foundation of China(11871257,12071130)。
文摘In this paper,we first establish the sharp growth theorem and the distortion theorem of the Frechet derivative for biholomorphic mappings defined on the unit ball of complex Banach spaces and the unit polydisk in C^(n) with some restricted conditions.We next give the distortion theorem of the Jacobi determinant for biholomorphic mappings defined on the unit ball of C^(n) with an arbitrary norm and the unit polydisk in C^(n) under certain restricted assumptions.Finally we obtain the sharp Goluzin type distortion theorem for biholomorphic mappings defined on the unit ball of complex Banach spaces and the unit polydisk in C^(n) with some additional conditions.The results derived all reduce to the corresponding classical results in one complex variable,and include some known results from the prior literature.
基金supported by NSF of China(11271359,11471098)Science and Technology Research Projects of Henan Provincial Education Department(17A110041)Scientific Research and Innovation Fund Projects of Zhoukou Normal University(ZKNUA201805)
文摘We mainly discuss the invariance of some subclasses of biholomorphic mappings under the generalized Roper-Suffridge operators on Bergman-Hartogs domains which are based on the unit ball Bn. Using the geometric properties and the distortion results of subclasses of biholomorphic mappings, we obtain the geometric characters of almost spirallike mappings of type β and order α, S_?~*(β, A, B), strong and almost spirallike mappings of type βand order α maintained under the generalized Roper-Suffridge operators on Bergman-Hartogs domains. Sequentially, we conclude that the generalized operators and the known operators preserve the same properties under some conditions. The conclusions generalize some known results.
基金This work was supported by 973 Project, the National Natural Science Foundation of China (Grant No. 19871081) the Natural Science Foundation of Guangdong Province and Anhui Province.
文摘The construction of normalized biholomorphic convex mappings in the Reinhardt domain $D_p = \{ (z_1 ,z_2 , \cdots ,z_n ) \in \mathbb{C}^n :\left| {z_1 } \right|^{p_1 } + \left| {z_2 } \right|^{p_2 } + \cdots + \left| {z_n } \right|^{p_n } < 1\} $ , p j > 2, j = 1,2,?, n) of ? n is discussed. The authors set up a Decomposition Theorem for such mappings. As a special case, it is proved that, for each such mapping f, the first k-terms of the homogeneous expansion of its j-th component f j , j = 1, 2, ?, n, depends only on z j , where k is the number that satisfies k < min {p 1, p 2,?, p n ≤ k + 1. When p1,p2, ... ,pn → ∞ , this derives the Decomposition Theorem of normalized biholomorphic convex mappings in the polydisc which was gotten by T.J. Suffridge in 1970.
基金Project supported in part by the National Natural Science Foundation of China.
文摘The power series expansions of normalized biholomorphic convex mappings on the Reinhardt domain are studied.It is proved that the first (k+1 ) terms ofthe expansions of the jth component fj of such a map / depend only on zj for 1≤j≤n,where k is the natural number that satisfies k<p≤k+1.When p∞,this gives the result on the unit polydisc obtained by Sulfridge in 1970.
文摘The estimations of the bounds of the Bloch constant of locally biholomorphic mappingson irreducible bounded symmetric domains are given. When the domain is a unit circle, theestimation of the lower bounds is just the famous one-half estimation.
基金Project supported by NSFC(Grant Nos.11271359 and 11471098)Science and Technology Research Projects of Henan Provincial Education Department(Grant Nos.17A110041 and 19B110016)Scientific Research Innovation Fund Project of Zhoukou Normal University(ZKNUA201805)
文摘In this paper, we generalize the Roper–Suffridge operator on the extended Hartogs domains.By using the geometric properties and the growth theorems of subclasses of biholomorphic mappings,we obtain the generalized operators preserve the properties of parabolic and spirallike mappings of type β and order ρ, S*_Ω(β, A, B), almost starlike mapping of complex order λ on ΩN under different conditions, and thus we get the corresponding results on the unit ball B^n in C^n. The conclusions lead to some known results.
基金supported by National Natural Science Foundation of China(Grant Nos.11501058 and 11431013)the Fundamental Research Funds for the Central Universities(Grant No.0208005202035)Key Research Program of Frontier Sciences,Chinese Academy of Sciences(Grant No.QYZDY-SSW-SYS001)
文摘Let Gi be a closed Lie subgroup of U（n）, Ωi be a bounded Gi-invariant domain in Cn which contains 0, and （9（Cn）Gi = C, for i= 1,2. If f ： f21 →2 is abiholomorphism, and f（0） = 0, then f is a polynomial mapping （see Ning et al. （2017））. In this paper, we provide an upper bound for the degree of such polynomial mappings. It is a natural generalization of the well-known Cartan＇s theorem.
文摘The distortion theorem for biholomorphic staxlike mappings(with respect to origin) inbounded symmetric domains are given.The distortion theorem for locally biholomorphic convexmappings in bounded symmetric domains are given also.
基金Project supported by the National Natural Science Foundation of China.
文摘1. In [1], T. J. Suffridge gave the necessary and sufficient condition for local biholomorphic mappings on the Reinhardt domains , to be starlike. Using this condition Yu Qihuang et al. gave the growth theorem and covering theorem for normalized biholomorphic starlike mappings in D_p. In this note we give a necessary and suffident condition for local biholomorphic mappings on more general Reinhardt
文摘The paper is made of two parts.In first part,We give the growth and 1/4-theorems for spiral like maps on the unit ball in l^p.Particularly,corresponding results were given in B^p.In the second part,we give the growth and 1/4-theorems for spirallike maps in an inner product space.We prove that the results is best.
基金supported by the National Natural Science Foundation of China(11001246,11101139)Zhejiang Innovation Project(T200905)
文摘In this note, the author introduces some new subcIasses of starlike mappings S^＊Ωn1p2,…,pn（β,A,B）={f∈H（Ω）：｜itanβ＋（1-itanβ）2/p（z）аp/аz（z）Jf^-1（z）f（z）-1-AB/1-B^2｜〈B-A/1-B^2},on Reinhardt domains Ωn1p2,…,pn=z∈C^n：｜z1｜^2＋n∑j=2｜zj｜^pj〈1}where - 1≤A〈B〈1,q=min{p2,…,pn}≥1,l=max{p2,…,pn}≥2 and β ∈（-π/2,π/2）.Some different conditions for P are established such that these classes are preserved under the following modified Roper-Suffridge operator F（z）=（f（z1）＋f＇（z1）Pm（z0）,（f＇（z1））^1/mz0）＇where f is a normalized biholomorphic function on the unit disc D, z = （z1,z0） ∈Ωn1p2,…,pn,z0=（z2,…,zn）∈ C^n-1.Another condition for P is also obtained such that the above generalized Roper-Suffridge operator preserves an almost spirallike function of type/3 and order β These results generalize the modified Roper-Suffridge extension oper-ator from the unit ball to Reinhardt domains. Notice that when p2 = p3 …=pn = 2,our results reduce to the recent results of Feng and Yu.
基金Project supported by National Natural Science Foundation of China(10971063,11061015)Major Program of Zhejiang Provincial Natural Science Foundation of China(D7080080)Guangdong Natural Science Foundation(06301315)
文摘In this article, first, a sufficient condition for a starlike mapping of order a f（x） defined on the unit ball in a complex Banach space is given. Second, the sharp estimate of the third homogeneous expansion for f is established as well, where f（z） = （f1（z）, f2（z）,..., fn（z））＇ is a starlike mapping of order a or a normalized biholomorphic starlike mapping defined on the unit polydisk in Cn, and D2fk（0）（z2） /2i= zk（∑l=1^b akzzl）, k = 2t l=1 k = 1, 2,..., n. Our result states that the Bieberbaeh conjecture in several complex variables （the case of the third homogeneous expansion for starlike mappings of order α and biholomorphic starlike mappings） is partly proved.
基金the NNSF of China(11671362,11971165)Beijing Municipal Natural Science Foundation(1182008)the Scientific Research Funds of Huaqiao University.
文摘This article is devoted to a deep study of the Roper-Suffridge extension operator with special geometric properties.First,we prove that the Roper-Suffridge extension operator preservesϵstarlikeness on the open unit ball of a complex Banach space C×X,where X is a complex Banach space.This result includes many known results.Secondly,by introducing a new class of almost boundary starlike mappings of orderαon the unit ball B n of C n,we prove that the Roper-Suffridge extension operator preserves almost boundary starlikeness of orderαon B n.Finally,we propose some problems.
文摘In this paper, we construct a new Roper-Suffridge extension operator Φn^r,β1,,βn（f）（z） = F（z） = （（rf（z1/r）/z1）^β1z1,（rf（z1/r）/z1）^β2z2,...,（rf（z1/r）/z1）^βnzn）＇,where f is a normalized locally biholomorphic function on the unit disc D, r = sup{｜z1｜ ： z =（z1, ···, zn） ∈Ω}, β1∈ [0, 1], 0 ≤βk≤β1, k = 2, ···, n, then we prove it can preserve the property of spirallikeness of type β, almost starlikeness of order α and starlikeness of orderα on bounded complete Reinhardt domain Ω, respectively.
基金partially supported by the NationalNatural Science Foundation of China(12071161,11971165,11701307)the Natural Science Foundation of Fujian Province(2020J01073)。
文摘This note is devoted to applying the principle of subordination in order to explore the Roper-Suffridge extension operator and the Pfaltzgraff-Suffridge extension operator with special analytic properties.First,we prove that both the Roper-Suffridge extension operator and the Pfaltzgraff-Suffridge extension operator preserve subordination.As applications,we obtain that if β∈[0,1],γ∈[0,1/r]and β+γ≤1,then the Roper-Suffridge extension operator Φ_(β+γ)(f)(z)=(f(z_(1)),(f(z_(1))/z_(1))^(β)(f’(z_(1)))^(γ)w),z∈Ω_(p,r) preserves an almost starlike mapping of complex order λ on Ω_(p,r)={z=(z_(1),w)∈C×X:|z_(1)|^(p)+‖w‖_(X)^(r)<1},where 1≤p≤2,r≥1 and X is a complex Banach space.Second,by applying the principle of subordination,we will prove that the Pfaltzgraff-Suffridge extension operator preserves an almost starlike mapping of complex order λ.Finally,we will obtain the lower bound of distortion theorems associated with the Roper-Suffridge extension operator.This subordination principle seems to be a new idea for dealing with the Loewner chain associated with the Roper-Suffridge extension operator,and enables us to generalize many known results from p=2 to 1≤p≤2.
基金This work was partially supported by the National Natural Science Foundation of China (Grant No.10471048)the Research Fund for the Doctoral Program of Higher Education (Grant No.20050574002)+1 种基金the Natural Science Foundation of Fujian Province of China (Grant No.Z0511013)the Education Commission Foundation of Fujian Province of China (Grant No.JB04038)
文摘The generalized Roper-Suffridge extension operatorΦ(f) on the bounded complete Rein- hardt domainΩin C^n with n≥2 is defined byΦ_(n,β_2,γ_2,...,β_n,γ_n)~r(f)(z)=rf((z_1)/r),((rf(z_1)/r)/(z_1))^(β2)(f′((z_1)/r)^(γ2)z_2,...,((rf(z_1)/r)/(z_1))^(β_n)(f′((z_1)/r))^(γ_n)z_n) for (z_1,z_2,...,z_n)∈Ω,where r=r(Ω) = sup{|z_1|:(z_1,z_2,...,z_n)∈Ω},0≤γ_j≤1-β_j,0≤β_j≤1, and we choose the branch of the power functions such that ((f(z_1))/(z_1))^(β_j)|_(z_1=0)=1 and (f′(z_1))^(γ_j)|_(z_1=0)= 1,j=2,...,n.In this paper,we prove that the operatorΦ_(n,β_2,γ_2,...,β_n,γ_n)~r(f) is from the subset of S_α~*(U) to Sα~*(Ω)(0≤α<1) onΩand the operatorΦ_(n,β_2,γ_2,...,β_n,γ_n)~r(f) preserves the starlikeness of order a or the spirallikeness of typeβon D_p for some suitable constantsβ_j,γ_j,p_j,where D_p= {(z_1,z_2,...,z_n)∈C^n:∑_(j=1)~n|z_j|^(p_j)<1}(p_j>0,j=1,2...,n),U is the unit disc in the complex plane C,and S_α~* (Ω) is the class of all normalized starlike mappings of orderαonΩ.We also obtain thatΦ_(n,β_2,γ_2,...,β_n,γ_n)~r(f)∈S_α~*(D_p) if and only if f∈S_α~*(U) for 0≤α<1 and some suitable constantsβ_j,γ_j,p_j.
