A NEW ROPER-SUFFRIDGE EXTENSION OPERATOR ON BOUNDED COMPLETE REINHARDT DOMAINS

In this paper, we construct a new Roper-Suffridge extension operator Φr n,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.     

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 GROWTH THEOREM FOR STARLIKE MAPPINGS ON BOUNDED STARLIKE CIRCULAR DOMAINS

1.IntroductionOnthegeometricfunctiontheoryofonecomplexvariable,thefollowinggrowthandi-coveringtheoremiswellknown(see[2]).TheoremA.Foreachno~alizedunivalentjunctionfontheunitdiscDCC,ESPecially,theleft-handsideOftheaboveinequalityimpliesf(D)2ID.ForeachzED,z/0,equalityoccursintheaboveinequalityifandonlyiffisKoe6efunctionK(z)=theoritsrotatione--"K(e"z).Itisnaturaltoextendthisandotherresultsonthegeometricfunctiontheoryofonevariablestoseveralvariables.Butasearlyasfiftyyearsago,H.Cartanpointe…     

THE EXTENSION OPERATOR IN BANACH SPACES FOR LOCALLY BIHOLOMORPHIC MAPPINGS

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.     

DISTORTION THEOREMS FOR SUBCLASSES OF STARLIKE MAPPINGS ALONG A UNIT DIRECTION IN C^n

In this paper, we obtain a distortion theorem of Jacobian matrix for biholomorphic subclasses of starlike mappings along a unit direction on the unit polydisc. These results extend the classical distortion theorem of starlike mappings to higher dimensions. We then give an upper bound estimate for biholomorphic subclasses of starlike mappings along a unit direction in a complex Banach space.     

THE PARAMETRIC REPRESENTATION FOR SPIRAL-LIKE MAPPINGS OF TYPE α ON BOUNDED BALANCED PSEUDOCONVEX DOMAINS

Let Ω be a bounded balanced pseudoconvex domain in Cn that admits a plurisubhamonic defining function of class C2. The authors prove that if f is a spiral-like mapping of type α on Ω, then f can be expressed as a limiting form concerning a Schwarz mapping. Moreover, the characteristics for star-like mappings and spiral-like mappings of type α are obtained. As a direct application, the growth theorem for spiral-like mappings of type α is set up.     

Loewner chains associated with the generalized Roper-Suffridge extension operator

In this paper, we introduce two classes of generalized Roper-Suffridge extension operators and prove that they can be embedded in Loewner chains. In particular, our proof shows that these two classes of operators preserve starlikeness and spirallikeness of type α on two important classes of Reinhardt domains in Cn, respectively. Finally, some other related results are given.     

齐型空间上的一类积分算子

在齐型空间上讨论由分数次积分算子、奇异积分算子及ＢＭＯ函数所构成的几类Ｔｏｅｐｌｉｔｚ型算子的有界性．     

Löwner chains and a subclass of biholomorphic mappings

In this paper, we give characterization of almost starlike functions of order α (respectively almost starlike mappings of order α) on the unit disc in C (respectively the unit ball in a finite-dimensional complex Banach space) in terms of Löwner chains. Furthermore, using the properties of Löwner chains, we can easily prove that two classes of generalized Roper-Suffridge extension operators preserve almost starlikeness of order α on two important classes of Reinhardt domains in Cn, respectively.     

AN INEQUALITY OF HOMOGENEOUS EXPANSION FOR BIHOLOMORPHIC QUASI-CONVEX MAPPINGS ON THE UNIT POLYDISK AND ITS APPLICATION

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.     

含有调和奇异算子的卷积型方程组的解法

讨论了二类离散的含有调和奇异算子的卷积型方程组,并通过离散的Laurent变换,把卷积型方程组化为具有间断系数的解析函数的Riemann边值问题,继而得到方程组的解.