The composition operators on weighted Bloch space

We will characterize the boundedness and compactness of the composition operators on weighted Bloch space B _log = { f ? H(D): sup_{z ? D} (1-| z|²) ( log\frac21-| z|² )| f￠(z)| B_{ \log }= \{ f \in H(D): \sup_{z \in D } (1-\left| z\right|^2) \left( \log \frac{2}{1-\left| z\right|^2} \right)\left| f'(z)\right| < +￥} +\infty \} , where H(D) be the class of all analytic functions on D.     

Carleson measure characterization of Bloch functions

We give several equivalences of Bloch functions and little Bloch functions. Using these results we obtain the generalized Carleson measure characterization of Bloch functions and the generalized vanishing Carleson measure characterization of little Bloch functions, that is,f B if and only if |D f(z)| ^p (1-|z|²)^p-1 dm(z) is a generalized Carleson measure;f B ₀ if and only if |D f(z)| ^p (1-|z|²)^p-1 dm(z) is a generalized vanishing Carleson measure, whereD f( > 0) is the fractional derivative of analytic functionf of order, m denotes the normalised Lebesgue measure.Supported partly by the Young Teacher Natural Science Foundation of Shandong Province.     

Univalent Functions, <Emphasis Type="Italic">VMOA</Emphasis> and Related Spaces

This paper is concerned mainly with the logarithmic Bloch space ℬ_log which consists of those functions f which are analytic in the unit disc \mathbbD{\mathbb{D}} and satisfy sup_{|z| < 1}(1-|z|)log\frac11-|z||f^￠(z)| < ￥\sup_{\vert z\vert <1}(1-\vert z\vert )\log\frac{1}{1-\vert z\vert}\vert f^{\prime}(z)\vert <\infty , and the analytic Besov spaces B ^p , 1≤p<∞. They are all subspaces of the space VMOA. We study the relation between these spaces, paying special attention to the membership of univalent functions in them. We give explicit examples of:

Bloch Functions in Several Complex Variables, I

Bloch functions on the unit disk are those analytic functionsf for which the quantity |f'(z)|(1 – |z|²) is bounded(for z in the disk). In this paper, Bloch functions on boundedhomogeneous domains in complex m-space are studied. The mainresult shows that many of the equivalent definitions of Blochfunctions on the unit disk are also equivalent in the generalsetting. Current Address: 39 Trinity College, Dublin 2, Ireland.     

Compact composition operators on the Bloch space in polydiscs

Let Uⁿ be the unit polydisc of ℂⁿ and φ=(φ₁, ⋯, φ _n ) a holomorphic self-map of Uⁿ. As the main result of the paper, it shows that the composition operator C is compact on the Bloch space β(Uⁿ) if and only if for every ε > 0, there exists a δ > 0, such that

Caratheodory inequality for analytic operator function

Abstract. Suppose H is a complex Hilbert space, AH (△) denotes the set of all analytic operator functions on     

Functions of bounded boundary rotation

Classes of functionsU _k, which generalize starlike functions in the same manner that the classV _k of functions with boundary rotation bounded bykπ generalizes convex functions, are defined. The radius of univalence and starlikeness is determined. The behavior off _α(z) = ∫ ₀ ^z [f'(t)]^α dt is determined for various classes of functions. It is shown that the image of |z|<1 underV _kfunctions contains the disc of radius 1/k centered at the origin, andV _k functions are continuous in |z|≦1 with the exception of at most [k/2+1] points on |z|=1.     

Some Extremal Functions in Fourier Analysis, III

We obtain the best approximation in L ¹(ℝ), by entire functions of exponential type, for a class of even functions that includes e ^−λ|x|, where λ>0, log |x| and |x| ^α , where −1<α<1. We also give periodic versions of these results where the approximating functions are trigonometric polynomials of bounded degree.     

Bloch型函数的高阶径向导数

讨论了复超球上全纯函数的高阶导数的增长速度 ,证明了f∈Bα 的充分必要条件是supa∈B(1- ｜z｜2 ) m+α- 1｜Rmf(z) ｜ <∞ ,或supa∈B∫B(1-｜z｜2 ) (m+α- 1) ｜Rmf(z) ｜pJRφα(z)dv(z) <∞ ,或 (1- ｜z｜2 ) p(m+α- 1) ｜Rmf(z) ｜pdv(z)是Bergman Carleson测度 .     

The Whittaker Function M κ ,μ as a Function of κ

The dependence of the Whittaker function M _{κ, μ} (z) on the parameter κ is considered. A convergent expansion in ascending powers and an asymptotic expansion in descending powers of κ are discussed. Some properties of the coefficients of the convergent expansion are shown. February 14, 1997. Date revised: March 5, 1998. Date accepted: March 19, 1998.     

Factorization for non-nevanlinna classes of analytic functions

A generalization of the Blaschke product is constructed. This product enables one to factor out the zeros of the members of certain non-Nevanlinna classes of functions analytic in the unit disc, so that the remaining (non-vanishing) functions still belong to the same class. This is done for the classesA ⁻ⁿ (0<n<∞) andB ⁻ⁿ (0<n<2) defined as follows:f ∈A ⁻ⁿ iff |f(z)|≦C _f (1−|z|)⁻ⁿ ,f ∈B ⁻ⁿ iff |f(z)|≦exp {C _f (1−|z|)⁻ⁿ }, whereC _f depends onf.     

On the Distribution Modulo 1 of q-Additive Functions

We consider the asymptotic behavior of the distribution functions defined by F_N(z)=¹―_N{1≦ n ≦ N : f(n)≦ z (mod 1)} in the case when f is q-additive. We give necessary and sufficient conditions for a q-additive function to have a uniform distribution modulo 1 or to have a non-uniform distribution modulo 1. This revised version was published online in June 2006 with corrections to the Cover Date.     

Composition operators on the Lipschitz space in polydiscs

Let Un be the unit polydisc of Cn and φ= (φ1,...,φn? a holomorphic self-map of Un. Let 0≤α< 1. This paper shows that the composition operator Cφ, is bounded on the Lipschitz space Lipa(Un) if and only if there exists M > 0 such thatfor z∈Un. Moreover Cφ is compact on Lipa(Un) if and only if Cφ is bounded on Lipa(Un) and for every ε > 0, there exists a δ > 0 such that whenever dist(φ(z),σUn) <δ     

Geometric conditions for interpolation in weighted spaces of entire functions

We use L² estimates for the equation to find geometric conditions on discrete interpolating varieties for weighted spaces A_p(ℂ) of entire functions such that |f(z)|≤Ae^Bp(z) for some A, B>0. In particular, we give a characterization when p(z)=e^|z| and more generally, when In p(e^r) is convex andIn p(r) is concave. Acknowledgements and Notes. The author wishes to thank X. Massaneda for useful talks and remarks.     

Characterization and the pre-Schwarzian norm estimate for concave univalent functions

Let Co(α) denote the class of concave univalent functions in the unit disk \mathbbD{\mathbb{D}}. Each function f ? Co(a){f\in Co(\alpha)} maps the unit disk \mathbbD{\mathbb{D}} onto the complement of an unbounded convex set. In this paper we find the exact disk of variability for the functional (1-|z|²)( f^￠￠(z)/f^￠(z)), f ? Co(a){(1-|z|^2)\left ( f^{\prime\prime}(z)/f^{\prime}(z)\right), f\in Co(\alpha)}. In particular, this gives sharp upper and lower estimates for the pre-Schwarzian norm of concave univalent functions. Next we obtain the set of variability of the functional (1-|z|²)(f^￠￠(z)/f^￠(z)), f ? Co(a){(1-|z|^2)\left(f^{\prime\prime}(z)/f^{\prime}(z)\right), f\in Co(\alpha)} whenever f′′(0) is fixed. We also give a characterization for concave functions in terms of Hadamard convolution. In addition to sharp coefficient inequalities, we prove that functions in Co(α) belong to the H ^p space for p < 1/α.     

Bloch-Bergman Pullbacks with Logarithmic Weights

We characterize the composition operators mapping Blochs boundedly into the weighted Bergman spaces of logarithmic weight. For 0 < p < ∞, 1 < α < ∞, let A^{p, log α} denote the space of holomorphic functions F in the unit disc D for which

Coefficients of nonvanishing functions inH ∞

LesB denote the class of functions analytic in the unit disc ofC which satisfy 0<|f(z)|<1. It is proved that there exists a numberc<1 such that iff∈B and iff(z)=Σ _n=0 ^∞ a _n z ⁿ , then |a _n |<c forn>=1.     

Estimates in the corona theorem and ideals ofH ∞ : A problem of T. Wolff

The main result of the paper is that there exist functionsf ₁,f ₂,f inH ^∞ satisfying the “corona condition”

Zero sets of holomorphic functions in the unit ball with slow growth

We study the zero-varieties of holomorphic functions in the unit ball satisfying the growth condition log |f(z)|≤c _fλ(|z|), where λ:(0,1)→ℝ⁺ is a positive increasing function. We obtain some sufficient conditions on an analytic variety to be defined by such a function. Some results for the particular case λ(r)=log(e/(1−r)), corresponding to the classA ^−∞, generalize those of B. Korenblum in one variable. Both authors supported by DGICYT grant PB92-0804-C02-02.     

Approximation by Faber-Laurent rational functions in the mean of functions of callsL p(Γ) with 1

Çavu§  A.  Israfilov  D. M. 《分析论及其应用》1995,11(1):105-118