In this paper,we extend the definition of almost starlike mappings of order 1/2 on B n to the almost starlike mappings of order α on the bounded starlike circular domains in C n,and give the growth theorem of it....In this paper,we extend the definition of almost starlike mappings of order 1/2 on B n to the almost starlike mappings of order α on the bounded starlike circular domains in C n,and give the growth theorem of it. This type of domain on which we discuss is rather general,in the sense that the domain must be starlike if there exists a normalized biholomorphic starlike mapping on it.展开更多
Suppose f is an almost starlike function of order α on the unit disk D. In this paper, we will prove that Фn,β2,γ2,…βn,γn（f）（z）=（f（z1）,（f（z1）/z1）^β2（f＇（z1））^γ2 z2,…,（f（z1）/z1）^βn（f＇（z...Suppose f is an almost starlike function of order α on the unit disk D. In this paper, we will prove that Фn,β2,γ2,…βn,γn（f）（z）=（f（z1）,（f（z1）/z1）^β2（f＇（z1））^γ2 z2,…,（f（z1）/z1）^βn（f＇（z1））^γnzn）1 preserves almost starlikeness of order α on Ωn,p2,…,pn={z=（z1,z2,…,zn）＇∈Cn：∑^n j=1｜zj｜^pj〈1},where 0〈p1≤2,pj≥1,j=2,…,n,are real numbers.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
This note induces some generalized Roper-Suffridge extension operators such that they are used to construct some almost starlike mappings of order α and starlike mappings of order a on different domains.
Let pj ∈ N and pj≥ 1, j = 2, …, k, k ≥ 2 be a fixed positive integer. We introduce a Roper-Suffridge extension operator on the following Reinhardt domain ΩN ={z =（z1, z′2,…, z′k）′∈ C × C^n2×…...Let pj ∈ N and pj≥ 1, j = 2, …, k, k ≥ 2 be a fixed positive integer. We introduce a Roper-Suffridge extension operator on the following Reinhardt domain ΩN ={z =（z1, z′2,…, z′k）′∈ C × C^n2×…× Cnk： ｜z1｜^2＋ ｜｜z2｜｜2^p2＋ … ＋ ｜｜zk ｜｜k^pk〈 1} given〈1} give by F（z）=（f（z1）＋f＇（z1）∑j=2 kPj（zj,（f＇（z1））1/p2 z2＇,…,（f＇（z1））1/pkzk＇）＇, where f is a normaljized biholomorphic function k（z） =（f（z1） ＋ f′（z1）=2 on the unit disc D, and for 2 ≤ j ≤ k, Pj ： C^nj→ C is a homogeneous polynomial of degree pj and zj =（zj1, …, zjnj）′∈ C^nj, nj ≥ 1, pj ≥ 1,｜｜zj｜｜j =（∑l=1 nj｜zjl｜^pj）1/pj. In this paper, some conditions for Pjare found under which the loperator p ｜zjl｜pj=1reserves the properties of almost starlikeness of order α, starlikeness of order αand strongly starlikeness of order α on ΩN, respectively.展开更多
A refining estimation of homogeneous expansion for / is discussed, where f belongs to a subclass of all normalized biholomorphic mappings defined on the unit polydisk in Cn or the unit ball in complex Banach spaces, a...A refining estimation of homogeneous expansion for / is discussed, where f belongs to a subclass of all normalized biholomorphic mappings defined on the unit polydisk in Cn or the unit ball in complex Banach spaces, and x = 0 is a zero of order k+1 of f(x)-x. Moreover, an estimation of homogeneous expansion for subordinate mappings defined on the unit ball in complex Banach spaces is also given.展开更多
文摘In this paper,we extend the definition of almost starlike mappings of order 1/2 on B n to the almost starlike mappings of order α on the bounded starlike circular domains in C n,and give the growth theorem of it. This type of domain on which we discuss is rather general,in the sense that the domain must be starlike if there exists a normalized biholomorphic starlike mapping on it.
基金Supported by the National Natural Science Foundation of China(10626015,10571044)Supported by the Guangdong Natural Science Foundation(06301315)+1 种基金Supported by the Doctoral Foundation of Zhanjiang Normal University(Z0420)Supported by the Natural Science Foundation of Henan University(XK03YBSX200)
文摘Suppose f is an almost starlike function of order α on the unit disk D. In this paper, we will prove that Фn,β2,γ2,…βn,γn（f）（z）=（f（z1）,（f（z1）/z1）^β2（f＇（z1））^γ2 z2,…,（f（z1）/z1）^βn（f＇（z1））^γnzn）1 preserves almost starlikeness of order α on Ωn,p2,…,pn={z=（z1,z2,…,zn）＇∈Cn：∑^n j=1｜zj｜^pj〈1},where 0〈p1≤2,pj≥1,j=2,…,n,are real numbers.
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Project supported by NNSF of China (10571164,10626015,10571044)SRFDP of Higher Education (20050358052)the Natural Science Foundation of Henan University (04ZDZR004).
文摘This note induces some generalized Roper-Suffridge extension operators such that they are used to construct some almost starlike mappings of order α and starlike mappings of order a on different domains.
基金Supported by the National Natural Science Foundation of China(11001074,11061015,11101124)
文摘Let pj ∈ N and pj≥ 1, j = 2, …, k, k ≥ 2 be a fixed positive integer. We introduce a Roper-Suffridge extension operator on the following Reinhardt domain ΩN ={z =（z1, z′2,…, z′k）′∈ C × C^n2×…× Cnk： ｜z1｜^2＋ ｜｜z2｜｜2^p2＋ … ＋ ｜｜zk ｜｜k^pk〈 1} given〈1} give by F（z）=（f（z1）＋f＇（z1）∑j=2 kPj（zj,（f＇（z1））1/p2 z2＇,…,（f＇（z1））1/pkzk＇）＇, where f is a normaljized biholomorphic function k（z） =（f（z1） ＋ f′（z1）=2 on the unit disc D, and for 2 ≤ j ≤ k, Pj ： C^nj→ C is a homogeneous polynomial of degree pj and zj =（zj1, …, zjnj）′∈ C^nj, nj ≥ 1, pj ≥ 1,｜｜zj｜｜j =（∑l=1 nj｜zjl｜^pj）1/pj. In this paper, some conditions for Pjare found under which the loperator p ｜zjl｜pj=1reserves the properties of almost starlikeness of order α, starlikeness of order αand strongly starlikeness of order α on ΩN, respectively.
基金This work was supported by the 973 Basic Research Program of China.
文摘A refining estimation of homogeneous expansion for / is discussed, where f belongs to a subclass of all normalized biholomorphic mappings defined on the unit polydisk in Cn or the unit ball in complex Banach spaces, and x = 0 is a zero of order k+1 of f(x)-x. Moreover, an estimation of homogeneous expansion for subordinate mappings defined on the unit ball in complex Banach spaces is also given.
