THE FEKETE-SZEG PROBLEM FOR CLOSE-TO-CONVEX FUNCTIONS WITH RESPECT TO THE KOEBE FUNCTION

An analytic function f in the unit disk D := {z ∈ C : |z| 1}, standardly normalized, is called close-to-convex with respect to the Koebe function k(z) := z/(1-z)2, z ∈ D, if there exists δ∈(-π/2, π/2) such that Re eiδ(1- z)2f′(z) 0, z ∈ D.For the class C(k) of all close-to-convex functions with respect to k, related to the class of functions convex in the positive direction of the imaginary axis, the Fekete-Szeg¨o problem is studied.     

Hardy-Sobolev类上的Bohr型不等式

Let β 〉 0 and Sβ ：= {z ∈ C ： ｜Imz｜ 〈β} be a strip in the complex plane. For an integer r ≥ 0, let H∞^Г,β denote those real-valued functions f on R, which are analytic in Sβ and satisfy the restriction ｜f^（r）（z）｜ ≤ 1, z ∈ Sβ. For σ 〉 0, denote by Bσ the class of functions f which have spectra in （-2πσ, 2πσ）. And let Bσ^⊥ be the class of functions f which have no spectrum in （-2πσ, 2πσ）. We prove an inequality of Bohr type
‖f‖∞≤π/√λ∧σ^r∑k=0^∞（-1）^k（r＋1）/（2k＋1）^rsinh（（2k＋1）2σβ）,f∈H∞^r,β∩B1/σ,
where λ∈（0,1）,∧and ∧′are the complete elliptic integrals of the first kind for the moduli λ and λ′=√1- λ^2,respectively,and λ satisfies
4∧β/π∧′=1/σ.
The constant in the above inequality is exact.     

THE RECOVERY OF FUNCTIONS OF PALEY-WIENER CLASS FROM IRREGULAR SAMPLINGS

It is shown that a function f which is in the classical Paley-Wiener class, and its k-th derivative f(k) can be recovered in the metric Lq(R),2 < q ≤ ∞, from its values on irregularly distributed discrete sampling set {tj}j∈z as limits of polynomial spline interpolation when the order of the splines goes to infinity, where {tj}j∈z is a real sequence such that {eifj(?)}j∈z constitutes a Riesz basis for L2([-π,π]).     

A Note on The Two-Functional Conjecture

Let D={z:|z|<1} and let K(D) denote the set of all functions analytic in D with the usual topology of uniform convergence on compact subsets of D. Let S be the class of function f(z) =z+a_2z~2+…analytic and univalent in D. Then S is a compact subset of K(D). A function f∈S is said to be a support point of S if it maximizes Re{L} over S for some continuous comp-     

On the Extension Operator in Banach Spaces

In [1], Roper and Suffridge introduced an extension operator. This operator is defined for normalized locally univalent function f on the unit disc U = {z∈C：｜z｜〈1} in C by Фn（f）（z）=（f（z1）,√ f′（z1）z0）, where z = （z1,z0） belongs to the unit ball Bn in Cn, z1∈U, z0 = （z2,...,zn） ∈ C^n-1.     

MODIFIED ROPER-SUFFRIDGE OPERATOR FOR SOME SUBCLASSES OF STARLIKE MAPPINGS ON REINHARDT DOMAINS

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.     

关于分担多项式的全纯函数的正规族(英文)

In this paper,we study normal families of holomorphic function concerning shared a polynomial.Let F be a family of holomorphic functions in a domain D,k（2）be a positive integer,K be a positive number andα（z）be a polynomial of degree p（p 1）.For each f∈F and z∈D,if f and f sharedα（z）CM and｜f（k）（z）｜K whenever f（z）-α（z）=0 in D, then F is normal in D.     

Differential inequalities, normality and quasi-normality

We prove that ifD is a domain in C,α 〉 1 and C 〉 0,then the family F of functions f meromorphic in D such that ｜f′（z）｜/1 ＋ ｜f（z）｜α 〉 C for every z ∈ D is normal in D.For α = 1,the same assumptions imply quasi-normality but not necessarily normality.     

单位球Bn上的Bohr不等式

Bohr＇s type inequalities are studied in this paper： if f is a holomorphic mapping from the unit ball B^n to B^n, f（0）=p, then we have sum from k=0 to∞｜DφP（P）[D^kf（0）（z^k）]｜/k!｜｜DφP（P）｜｜〈1 for｜z｜〈max{1/2＋｜P｜,（1-｜p｜）/2^1/2andφ_P ∈Aut（B^n） such thatφ_（p）=0. As corollaries of the above estimate, we obtain some sharp Bohr＇s type modulus inequalities. In particular, when n=1 and ｜P｜→1, then our theorem reduces to a classical result of Bohr.     

APPROXIMATION BY FABER-LAURENT RATIONAL FUNCTIONS IN THE MEAN OF FUNCTIONS OF CLASS LP(T) WITH 1

A. Cavus  D. M. Israfilov 《分析论及其应用》1995,(1)

关于亚纯函数分担值集正规族的注记(英文)

In the paper,we prove the main result：Let k（≥2）be an integer,and a,b and c be three distinct complex numbers.Let F be a family of functions holomorphic in a domain D in complex plane,all of whose zeros have multiplicity at least k.Suppose that for each f∈F,f（z）and f（k）（z）share the set{a,b,c}.Then F is a normal family in D.     

Some Theorems of Analytic Functions

1 Introduction We denote that: σ—the class of functions ω(z)=A_1z+A_2z~2+…regular in the unit disk such that sum from n=1 to ∞ (n|A_n|~2<∞);K_c— the class of close-to-convex function f(z),that is, if f(z)=α_1z+α_2z~2+…there exists a starlike function g(z) =b_1z+b_2z~2+…such that     

Periodic points and normal families concerning multiplicity

In 1992,Yang Lo posed the following problem:let F be a family of entire functions,let D be a domain in C,and let k 2 be a positive integer.If,for every f∈F,both f and its iteration f~khave no fixed points in D,is F normal in D?This problem was solved by Ess′en and Wu in 1998,and then solved for meromorphic functions by Chang and Fang in 2005.In this paper,we study the problem in which f and f~(k ) have fixed points.We give positive answers for holomorphic and meromorphic functions.(I)Let F be a family of holomorphic functions in a domain D and let k 2 be a positive integer.If,for each f∈F,all zeros of f(z)-z are multiple and f~khas at most k distinct fixed points in D,then F is normal in D.Examples show that the conditions"all zeros of f(z)-z are multiple"and"f~k having at most k distinct fixed points in D"are the best possible.(II)Let F be a family of meromorphic functions in a domain D,and let k 2 and l be two positive integers satisfying l 4 for k=2 and l 3 for k 3.If,for each f∈F,all zeros of f(z)-z have a multiplicity at least l and f~khas at most one fixed point in D,then F is normal in D.Examples show that the conditions"l 3for k 3"and"f~k having at most one fixed point in D"are the best possible.     

Jackson＇s Theorem on Bounded Symmetric Domains

Polynomial approximation is studied on bounded symmetric domain Ω in C^n for holomorphic function spaces X such as Bloch-type spaces, Bergman-type spaces, Hardy spaces, Ω algebra and Lipschitz space. We extend the classical Jackson＇s theorem to several complex variables：Eκ（f,X）≤ω（1/k,f,X）, where Eκ（f,X） is the deviation of the best approximation of f ∈X by polynomials of degree at most k with respect to the X-metric and ω（1/k,f,X） is the corresponding modulus of continuity.     

THE SPHERICAL FUNCTIONS AND THE CENTRAL LIMIT THEOREM ON SU（2,2）/S（U（2） × U（2））

Let G = SU（2, 2）, K = S（U（2） × U（2））, and for l ∈ Z, let {Tl}l∈z be a one-dimensional K-type and let El be the line bundle over G/K associated to Tl. It is shown that the Tl-spherical function on G is given by the hypergeometric functions of several variables. By applying this result, a central limit theorem for the space G/K is obtained.     

Area Estimates of Images of Disks under Analytic Functions

Let D_r := {z = x + iy ∈ C : |z| r}, r ≤ 1. For a normalized analytic function f in the unit disk D := D1, estimating the Dirichlet integral Δ(r, f) =∫∫_(D_r)|f'(z)|~2 dxdy, z = x + iy, is an important classical problem in complex analysis. Geometrically, Δ(r, f) represents the area of the image of D_r under f counting multiplicities. In this paper, our main ob jective is to estimate areas of images of D_r under non-vanishing analytic functions of the form(z/f)~μ, μ 0, in principal powers,when f ranges over certain classes of analytic and univalent functions in D.     

Convergence of Gaussian Quadrature Formulas for Power Orthogonal Polynomials

In classical theorems on the convergence of Gaussian quadrature formulas for power orthogonal polynomials with respect to a weight w on I =（a,b）,a function G ∈ S（w）:= { f:∫_I | f（x）| w（x）d x < ∞} satisfying the conditions G ^2j（x） ≥ 0,x ∈（a,b）,j = 0,1,...,and growing as fast as possible as x → a + and x → b,plays an important role.But to find such a function G is often difficult and complicated.This implies that to prove convergence of Gaussian quadrature formulas,it is enough to find a function G ∈ S（w） with G ≥ 0 satisfying sup n ∑λ0knG（xkn） k=1 n<∞ instead,where the xkn ’s are the zeros of the n th power orthogonal polynomial with respect to the weight w and λ0kn ’s are the corresponding Cotes numbers.Furthermore,some results of the convergence for Gaussian quadrature formulas involving the above condition are given.     

Distortion theorem for Bloch mappings on the unit ball <Emphasis Type="Italic">ℬ</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In this paper, we obtain a version of subordination lemma for hyperbolic disk relative to hyperbolic geometry on the unit disk D. This subordination lemma yields the distortion theorem for Bloch mappings f ∈ H（B^n） satisfying ｜｜f｜｜0 = 1 and det f＇（0） = α ∈ （0, 1], where｜｜f｜｜0 = sup{（1 - ｜z｜^2 ）n＋1/2n det（f＇（z））[1/n ： z ∈ B^n}. Here we establish the distortion theorem from a unified perspective and generalize some known results. This distortion theorem enables us to obtain a lower bound for the radius of the largest univalent ball in the image of f centered at f（0）. When a = 1, the lower bound reduces to that of Bloch constant found by Liu. When n = 1, our distortion theorem coincides with that of Bonk, Minda and Yanagihara.     

Normal Functions Concerning Shared Values

In this paper we discuss normal functions concerning shared values. We obtain the follow result. Let F be a family of meromorphic functions in the unit disc A, and a be a nonzero finite complex number. If for any f ∈F, the zeros of f are of multiplicity, f and f′ share a, then there exists a positive number M such that for any f∈F1（1-｜z｜^2） ｜f′（z）｜/1＋｜f（z）｜^2≤M.     

Bers型空间和复合算子

For α∈（0,∞),let Hα^∞（or Hα^∞,0)denote the collection of all functions f which are analytic on the unit disc D and satisfy |f(z)|(1-|z|^2)^α=O(1)(or|f(z)|(1-|z|^2)^α=o(1) as |z|→1).Hα^∞,0)is called a Bers-type space (or a little Bers-type space).In this paper,we give some basic properties of Hα^∞，Cψ，the composition operator associated with a symbol function ψ which is an analytic self map of D,is difined by Cψf=f o ψ,We characterize the boundedness,and compactness of Cψ which sends one Bers-type space to another function space.