多复变一类星形映照子族的系数不等式（英文）

Let S~* be the familiar class of normalized univalent functions in the unit disk.In [9], Keogh and Merkes proved that for a function f(z) = z +∑k=2∞ a_kz~k in the class S~*,then |a_3-λa_2~2| ≤ max{1, |3-4λ|}, λ∈ C. In this article, we investigate the corresponding problem for the subclass of starlike mappings defined on the unit ball in a complex Banach space, on the unit polydisk in Cnand the bounded starlike circular domain in C~■, respectively.     

Sharp Growth Theorems and Coefficient Bounds for Starlike Mappings in Several Complex Variables

Let B be the unit ball in a complex Banach space. Let S^＊k＋1（B） be the family of normalized starlike mappings f on B such that z = 0 is a zero of order k ＋ 1 of f（z） - z. The authors obtain sharp growth and covering theorems, as well as sharp coefficient bounds for various subsets of S^＊k＋1（B）.     

ON THE SHARP GROWTH, COVERING THEOREMS FOR NORMALIZED BIHOLOMORPHIC MAPPINGS IN C^n

In this article.a normalized biholomorphic mapping f defined on bounded starlike circular domain in C″is considered,where z=0 is a zero of order k 1 of f(z)(?) The sharp growth.covering theorems for almost starlike mappings of order a and starlile mappings of order wave established.Meanwhile,the construction of the above mappings on bounded starlike circular domain in C~n is also discussed,it provides the extremal mappings for the growth,covering theorems of the above mappings.     

Non-wandering points and the depth for graph maps

Let G be a graph and f:G→G be continuous.Denote by R(f) andΩ(f) the set of recurrent points and the set of non-wandering points of f respectively.LetΩ_0(f) = G andΩ_n(f)=Ω(f|_(Ω_(n-1)(f))) for all n∈N.The minimal m∈NU {∞} such thatΩ_m(f)=Ω_(m 1)(f) is called the depth of f.In this paper,we show thatΩ_2 (f)=(?) and the depth of f is at most 2.Furthermore,we obtain some properties of non-wandering points of f.     

A normal criterion about two families of meromorphic functions concerning shared values

In this paper, we mainly discuss the normality of two families of functions concerning shared values and proved: Let F and G be two families of functions meromorphic on a domain D■C,a1, a2, a3, a4 be four distinct finite complex numbers. If G is normal, and for every f ∈ F , there exists g ∈ G such that f(z) and g(z) share the values a1, a2, a3, a4, then F is normal on D.     

The generalized Roper-Suffridge extension operator on bounded complete Reinhardt domains

The generalized Roper-Suffridge extension operator Ф(f) on the bounded complete Reinhardt domain Ω in Cn with n ≥ 2 is defined by Φrn,β2,γ2,…,βn,γn(f)(z)=(rf(z1/r),(rf(z1/r)/z1)β2(f'(z1/r))γ2z2,…,(rf(z1/r)/z1)βn(f'(z1/r)γnzn) for (z1,z2,…,zn) ∈Ω, where r = r(Ω) = sup{|z1| (z1,z2,…,zn) ∈ Ω},0 ≤ γj ≤ 1 -βj,0 ≤ βj ≤ 1,and we choose the branch of the power functions such that (f(z1)/z1)βj |z1=0 = 1 and (f′(z1))γj |z1=0 =1,j = 2,…,n. In this paper, we prove that the operator Фrn,β2,γ2,…,βn,γn(f) is from the subset of S*α(U) to S*α(Ω)(0 ≤ α ＜ 1) on Ω and the operator Фrn,β2,γ2,…, βn,γn(f) preserves the starlikeness of order α or the spirallikeness of type β on Dp for some suitable constantsβj,γj,pj, where Dp ={(z1,z2,…,zn) ∈ Cn ∑nj=1|zj|pj ＜ 1} (pj ＞ 0, j = 1,2,…,n), U is the unit disc in the complex plane C, and Sα* (Ω) is the class of all normalized starlike mappings of order α on Ω. We also obtain that Φrn,β2,γ2,…,γn(f) ∈ S*α(Dp) if and only if f ∈ S*a(U) for 0 ≤ α ＜ 1 and some suitable constants βj,γj,pj.     

图映射的非游荡集

§ 1　IntroductionLet N be the set of all natural numbers.Write Z+=N∪ { 0 } ,Nn={ 1 ,2 ,...,n} andZn={ 0 }∪Nnfor any n∈N.Let X be a topological space and f:X→X be a continuous map.Forx∈X,O(x,f) ={ fk(x) :k∈ Z+} is called the orbit of x.The set of periodic points,the set of recurrentpoints,the set ofω-limit points for some x∈X and the set of non-wandering points of fare denoted by P(f) ,R(f) ,ω(x,f) andΩ(f) ,respectively(for the definitions see[1 ] ) .Let A X,we use int(A) ,A…     

空间中渐近非扩张型映射的渐近行为

Let X be a uniformly convex Banach space X such that its dual X^＊ has the KK property. Let C be a nonempty bounded closed convex subset of X and G be a directed system. Let ={Tt ： t ∈ G} be a family of asymptotically nonexpansive type mappings on C. In this paper, we investigate the asymptotic behavior of {Ttx0 ： t∈ G} and give its weak convergence theorem.     

Some properties for certain classes of univalent functions defined by differential inequalities

Let A be the space of functions analytic in the unit disk D = {z:|z| 1}.Let U denote the set of all functions f ∈ A satisfying the conditions f(0) = f'(0)-1 = 0 and|f'(z)(z/f(z))~2-1|1(|z|1).Also,let Ω denote the set of all functions f ∈ A satisfying the conditions f(0) = f'(0)-1 = 0and|zf'(z)-f(z)|1/2(|z|1).In this article,we discuss the properties of U and Ω.     

Meromorphic Convex Functions with Negative Coefficients

Let σ_k(a) be the class of functions f(f)=1/z-sur from n=1 to ∞(|a_n|z~n), regular in the punctured disk E={z:0<|z|<1} and satisfying Re(1 zf"(z)/f'(z))<-a (0≤a<1) for z∈E. In this paper we obtain coefficient inequalities, distortion and closure Theorems for the class σ_k(a). Further we obtain the class preserving integral operator of the form     

ROPER-SUFFRIDGE EXTENSION OPERATOR ON A REINHARDT DOMAIN

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 × Cn2×···× Cnk: |z1|2+ ||z2||p22+ ··· + ||zk ||pk k 1} given11 by F P′j(zj),(f(z1))p2 z′2, ···,(f′(z1))pk z′k)′, 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 : Cnj-→ C is a homogeneous polynomial of degree pj and zj =(zj1, ···, zjnj)′∈ Cnj, nj ≥ 1, pj ≥ 1,nj1||zj ||j =()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.     

THE SHARP ESTIMATE OF THE THIRD HOMOGENEOUS EXPANSION FOR A CLASS OF STARLIKE MAPPINGS OF ORDER α ON THE UNIT POLYDISK IN Cn

In this article,first,a sufficient condition for a starlike mapping of order α f(x) defined on the unit ball in a complex Banach space is given. Second,the sharp estimate of the third homogeneous expan...     

α次的殆星映射的充分判别条件

该文通过在cⁿ中的有界星形圆型域和复Banach空间中的单位球上建立一些偏微分不等式, 分别给出α 次的殆星映射在这两类域上的充分判别条件.     

多复变Fekete-Szeg问题

令y为单位圆盘上正规化单叶函数族.Fekete和Szeg证明了如下的著名结果:对λ∈[0,1],成立max_(f∈y)|a_3-λa_2~2|=1+2~(-(2λ)/(1-λ)).本文研究了C~n中有界星形圆形域上的星形映照的相应问题.定理的证明有假设条件,但该条件在单复变情形下是自动满足的.     

Radius of Starlikeness for Class S(a,n)and Its Extension

This paper obtain that the radius of starlikeness for class S(α,n)in [1] is,tespectivety,where α_ is unique solution of equation (αα)~(1/2)=σwith a in (0.1),and α-[1+(1-2α)r~(2n)]/(1-r~(2n)),σ=[1-(1-2α)r~]/(1+r~).Futhermore,we consider an extension of class S(α,n):Let S(α、β、n)denote the class of functions f(z)=z+α_z~(n+1)+…(n≥1)that are analytie in |z|<1 such that f(z)/g(z)∈p(α,n)[1],where g(z)∈S~*(β)[2].This paper prove that the radius of starlikeness of class S(α,β,n) is given by the smallest positive root(less than 1)of the following equations(1-2α)(1-2β)r~(2)-2[1-α-β-n(1-α)]r~+1=0.0≤α≤α_0,(1-α)[1-(1-2β)r~]-n[r~(1+r~)=0.,α_0≤α<1.where α=[1+(1-2α)r~(2)]/(1-r~(2)(0≤r<1),α_0(?(0,1) is some fixed number.This result is also thecxtension of well-known results[T.Th3] and [8,Th3]     

在Banach空间中推广的Roper-Suffridge算子(Ⅰ)

本文将Cn中的Roper-Suffridge算子推广到任意复Banach空间中,并证明这种算子在任意复Banach空间中的某些区域上具有保持ε星形性,由此可以构造出任意复Banach空间,复Hilbert空间和Cn中的一些区域上的许多双全纯星形映照、双全纯凸映照、双全纯ε星形映照,同时,得到它们的增长定理等,将龚升与刘太顺,Roper与suffridge,Graham,Kohr等学者在Cn中的一些结果推广到任意复Banach空间或复Hilbert空间中．     

A refinement about estimation of expansion coefficients for normalized biholomorphic mappings

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.     

The Covering Theorems for Normalized Holomorphic Mappings on Bounded Starlike Circular Domains in C~n

In this paper, we will investigate convex domains and starlike domains which contain the image set of normalized holomorphic mappings on bounded starlike circular domains in Cn.