A p Interpolation in the Unit Ball

We study interpolating sequences in the unit ball for A^–pwith p > 0, the Banach space of holomorphic functions f with(1 – |z|²)^p |f(z)| bounded. The finite unions of A^–p-interpolatingsequences are characterized by a Carleson type condition.     

Sharp Distortion Estimate for Locally Schlicht Bloch Functions

Let B be the space of locally schlicht Bloch functions f whichare analytic in the unit disc with f(0) = f'(0) – 1 =0 satisfying 0 < |f'(z)|(1 – |z|²) 1. For each fixedz₀ we shall determine the shape of the set {logf'(z₀): fB},that is, we shall give the sharp distortion estimate for locallyschilcht Bloch functions.     

Multipliers of Qs spaces and the corona theorem

For s>0, let Q_s be the space of all analytic functions onthe unit disc such that |f'(z)|²(1–|z|²)^s dA(z) isan s-Carleson measure. Here we prove that the corona theoremholds for the algebra of pointwise multipliers of Q_s.     

Harmonic Analogues of G. R. Maclane's Universal Functions

Let E denote the space of all entire functions, equipped withthe topology of local uniform convergence (the compact-opentopology). MacLane [15] constructed an entire function f whosesequence of derivatives (f, f', f', ...) is dense in E; hisconstruction is succinctly presented in a much later note byBlair and Rubel [2], who unwittingly rederived it (see also[3]). We shall call such a function f a universal entire function.In this note we show that analogous universal functions existin the space H_N of functions harmonic on R^N, where N2. We alsostudy the permissible growth rates of universal functions inH_N and show that the set of all such functions is very large. For purposes of comparison, we first review relevant facts aboutuniversal entire functions. The function constructed by MacLaneis of exponential type 1. Duyos Ruiz [7] observed that a universalentire function cannot be of exponential type less than 1. G.Herzog [11] refined MacLane's growth estimate by proving theexistence of a universal entire function f such that |f(z)|=O(re^r)as |z|=r. Finally, Grosse–Erdmann [10] proved the followingsharp result.     

Distortion theorems for Bloch mappings on the unit polydisc D

In this article, we establish distortion theorems for some various subfamilies of Bloch mappings defined in the unit polydisc Dⁿ with critical points, which extend the results of Liu and Minda to higher dimensions. We obtain lower bounds of |det (f'(z))| and ? det (f'(z)) for Bloch mapping f. As an application, some lower and upper bounds of Bloch constants for the subfamilies of holomorphic mappings are given.     

A Counterexample to Uniqueness in the Riemann Mapping Theorem for Univalent Harmonic Mappings

Let f be an orientation-preserving univalent harmonic mappingof the unit disk U. Then , where h and g are analytic in U. Furthermore, f satisfies theequation in U, where a(z)=g'(z)/h'(z), and |a(z)| < 1 in U. The functiona(z) is the analytic dilatation of f. In [2], Hengartner and Schober proved the following versionof the Riemann mapping theorem for univalent harmonic mappings.1991 Mathematics Subject Classification 31A05, 31A20.     

De Paepe's Disc has Nontrivial Polynomial Hull

The topological disc (De Paepe's) isshown here to have non-trivial polynomially convex hull. Infact, the authors show that this holds for all discs of theform , where f is holomorphicon |z|r, and f(z=z²+a₃z³+..., with all coefficients a_n real,and at least one a_2n+1 0. 2000 Mathematics Subject Classification32E20.     

On typically real functions which are generated by a fixed typically real function

Let T be the family of all typically real functions, i.e. functions that are analytic in the unit disk Δ:= {z ∈ ℂ: |z| < 1}, normalized by f(0) = f′(0) − 1 = 0 and such that Imz Im f(z) ⩾ 0 for z ∈ Δ.     

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.     

一类算子值解析函数族的极值点

设 H 是一个Hilbert空间. B(H) 表示所有H 到 H 的有界线性算子构成的Banach空间. 设 T= {f(z): f(z)=zI-∑^∞_n=2 zⁿA_n 在单位圆盘|z|<1上解析, 其中系数A_n是 H 到 H 的紧正Hermitian算子, I 表示 H 上的恒等算子, ∑^∞_n=2 n(A_n x, x) ≤1 对所有x ∈H, ∣|x∣∣=1 成立. 该文研究了函数族 T 的极值点.     

Interpolating Blaschke Products and Factorization Theorems

Let M(H) be the maximal ideal space of H the Banach algebraof bounded analytic functions on the open unit disk. Let G bethe set of nontrivial points in M(H). By Hoffman's work, G hasdeep connections with the zero sets of interpolating Blaschkeproducts. It is proved that for a closed -separated subset Eof M(H) with E G, there exists an interpolating Blaschke productwhose zero set contains E. This is a generalization of Lingenberg'stheorem. Let f be a continuous function on M(H). Suppose thatf is analytic on a nontrivial Gleason part P(x), f(x) = 0, andf 0 on P(x). It is proved that there is an interpolating Blaschkeproduct b with zeros {z_n}_n such that b(x) = 0 and f(z_n) = 0for every n. This fact can be used for factorization theoremsin Douglas algebras and in algebras of functions analytic onGleason parts.     

Finely Harmonic Functions need not be Quasi-Analytic

If a function f is harmonic on a connected open set R^d andis constant in a neighbourhood of one point in then it is identicallyconstant. We give an example of a non-constant finely harmonicfunction defined on E={zC||z|<2}\p which is identically zeroon |z|1. The exceptional set P is of capacity zero so E is finelyopen and finely connected. This example therefore shows thatfinely harmonic functions are not globally determined by theirlocal behaviour.     

Pad?-approximant Bounds and Approximate Solutions for Kirkwood-Riseman Integral Equations

Two-point Pad? approximants are used to calculate tight upperand lower bounds on the quantity <?, f> associated withKirkwood-Riseman integral equations (1+yL)?=f, which arise inthe diffusion theory of flexible macromolecules. The self-adjointoperator L is an integral operator on –1 x 1, with weaklysingular kernel |x – x'|^–?, and the two specificcases (i) f = 1, (ii) f = x² are studied. In case (i) directbounds on <?, 1> are obtained; this quantity is inverselyproportional to the translational diffusion constant. In case(ii) bounds on <?, 1 > are found by a new technique involvingcombinations of bounds for the three cases f = 1, f = x² andf = bx²?b^–1. Various types of Pade and related approximantsare compared, using the information <f, Lⁿf>, n = –2,–1, 0, 1, 2, 3 and (an upper bound on L) for severalvalues of the positive parameter y. Pad?-approximant-generating trial vectors are investigated anda convergence theorem is established. The vector consistingof an optimum linear combination of L^–1f, f and Lf isfound to be an accurate approximation to a numerical solutionin case (ii), for all values of y and x. Specific analyticalexpressions are derived for the approximate solutions.     

Regions of Values of the Systems {f(z 1), f(z 2), f'(z 2)} and {f(z 1), f'(z 1), f'(z 1)} on the Class of Typically Real Functions

Let T be the class of functions f(z) = z + a ₂ z ² + . . . that are regular in the unit disk and satisfy the condition Im f(z) Im z > 0 for Im z 0, and let z ₁ and z ₂ be any distinct fixed points in the disk |z| < 1. For the systems of functionals mentioned in the title, the regions of values on T are studied. As a corollary, the regions of values of f'(z ₂) and f'(z ₁) on the subclasses of functions in T with fixed values f (z ₁), f (z ₂) and f (z ₁), f'(z ₁), respectively, are found. Bibliography: 7 titles.     

On Weighted Distortion in Conformal Mapping II

Some years ago, Blatter [1] gave a result of the form for any function f regular and univalentin D: |z| < 1, where is the hyperbolic distance betweenz₁ and z₂. Kim and Minda [5] pointed out that the multiplieron the right is incorrect. They say that Blatter's proof givesthe correct multiplier, but Blatter's formula for in termsof z₁, z₂ is wrong. Kim and Minda proved the generalized formula where D₁(f) = f'(z) (1 – |z|²),valid for p P with some P, . In each case there was an appropriate equality statement. Kimand Minda made the important and easily verified remark thatthese problems are linearly invariant in the sense that if theresult is proved for f, then it follows for , where U is a linear transformation of the planeonto itself and T is a linear transformation of D onto itself.This means that we need to prove such results only in an appropriatelynormalized context. 1991 Mathematics Subject Classification30C75, 30F30.     

A coefficient inequality for Bloch functions with applications to uniformly locally univalent functions

We give a Fekete-Szeg? type inequality for an analytic function on the unit disk with Bloch seminorm ≤1. As an application of it, we derive a sharp inequality for the third coefficient of a uniformly locally univalent function f(z) = z + a ₂ z ² + a ₃ z ³ + ⋯ on the unit disk with pre-Schwarzian norm ≤λ for a given λ > 0.     

On power deformations of univalent functions

For an analytic function f (z) on the unit disk |z| < 1 with f (0) = f′(0) − 1 = 0 and f (z) ≠ 0, 0 < |z| < 1, we consider the power deformation f _c (z) = z(f (z)/z) ^c for a complex number c. We determine those values c for which the operator maps a specified class of univalent functions into the class of univalent functions. A little surprisingly, we will see that the set is described by the variability region of the quantity zf′(z)/ f (z), |z| < 1, for most of the classes that we consider in the present paper. As an unexpected by-product, we show boundedness of strongly spirallike functions.     

The Existence of Periodic and Subharmonic Solutions of Subquadratic Second Order Difference Equations

By critical point theory, a new approach is provided to studythe existence of periodic and subharmonic solutions of the secondorder difference equation where f C(R x R^m, R^m), f(t+M,z)+f(t,z) for any (t, z)R x R^mand M is a positive integer. This is probably the first timecritical point theory has been applied to deal with the existenceof periodic solutions of difference systems.     

Mobius Instability of Sampling for Small Weighted Spaces of Analytic Functions

In this paper, the space A (D)is considered, consisting of thoseholomorphic functions f on the unit disk D such that || f ||= sup_{z D} | f(z)|(|z|) < +, with (1) = 0. The sampling problemis studied for weights satisfying ln (r)/ln(1 – r) 0.Möbius stability of sampling is shown to fail in this space.2000 Mathematics Subject Classification 30H05 (primary), 30D60(secondary).     

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.