Computational Methods and Function Theory

Muir, Stacey

2006 Computational Methods and Function Theory

doi: 10.1007/BF03321627

In 2003, Ruscheweyh and Suffridge settled a conjecture of Pólya and Schoenberg on subordination of the de la Vallée Poussin means of an analytic function by defining a continuous extension of the de la Vallée Poussin means using a differential equation. We extend this differential equation to a more general setting and observe that a similar subordination result with convex functions holds. Through an integral operator of Bernardi, particular convex subordination chains are constructed with specified limiting functions. Finally, we show the importance of convexity by producing an example of a family of starlike solutions that fails to be a subordination chain.

Katkova, Olga

2006 Computational Methods and Function Theory

doi: 10.1007/BF03321628

This paper investigates the relationship between the Riemann ζ-function and various classes of generating functions of multiply positive sequences introduced by Schoenberg.

Batra, Prashant

2006 Computational Methods and Function Theory

doi: 10.1007/BF03321629

Gol’dberg considered the class of functions analytic in the unit disc with unequal positive numbers of zeros and ones there. The maximum modulus of zero- and one-places in this class is non-trivially bounded from below by the universal constant A 2. This constant determines a fundamental limit of controller design in engineering, and has applications when estimating covering regions for composites of fixed point free functions with schlicht functions. The lower bound for A 2 is improved in this note by considering simultaneously the extremal functions f and 1 — f together with their reciprocals.

Hjelle, Geir

2006 Computational Methods and Function Theory

doi: 10.1007/BF03321630

We give a constructive and flexible proof of a result of P. Gorkin and R. Mortini concerning a special finite interpolation problem on the unit circle with interpolating Blaschke products. Our proof also shows that the result can be generalized to closed curves other than the unit circle.

Polyakov, Peter

2006 Computational Methods and Function Theory

doi: 10.1007/BF03321631

We study solvability of the generalized Possio integral equation which is a tool in the analysis of a boundary value problem in 2D subsonic aeroelasticity with the Kutta-Joukowski condition — “zero pressure discontinuity” — ψ(x, 0, t) = 0 on the complement of a finite interval in the entire real line ℝ. The corresponding problem with boundary condition on finite intervals adjacent to the “chord” was considered in [16].

Akeroyd, John

2006 Computational Methods and Function Theory

doi: 10.1007/BF03321633

Let Ω be a subregion of {z : |z| < 1} for which the Dirichlet problem is solvable, assume that 0 ∈ Ω and let ω Ω denote harmonic measure on ∂Ω for evaluation at 0. If E is a Borel subset of {z : |z| = 1} and ω Ω (E) > 0, then we find a simply connected region G, where 0 ∈ G ⊆ {z : |z| < 1}, ∂G ⊆ Ω ∪ E and ω G (E) > 0, such that U := G ∪ Ω has the property that w U and w Ω are boundedly equivalent on ∂U. We mention consequences of this in function theory.

Avkhadiev, Farit; Wirths, Karl-Joachim

2006 Computational Methods and Function Theory

doi: 10.1007/BF03321634

Let D denote the open unit disc and f: D → ℂ be holomorphic and injective in D such that f(D) is a convex domain and f(0) = f’(0) − 1 = 0. Let F be the inverse function of f defined in a neighbourhood of the origin and k ∈ ℕ. We consider the Taylor expansions $$(F(w))^k=\sum_{n=k}^{\infty} A_{n,k}w^{n}$$ . We prove that the inequality $$\biggr|\sum_{k=1}^nA_{n,k}\biggr| \leq 2^{n-1}$$ is valid for any n ∈ ℕ and that equality occurs in this inequality for a fixed n ≥ 2 if and only if f(z) = z/(1 + z).

Matheson, Alec; Ross, William

2006 Computational Methods and Function Theory

doi: 10.1007/BF03321635

A classical theorem of Frostman says that if B is a Blaschke product (or any inner function), then its Frostman shifts $$B_w = \left( {B - w} \right)\left( {1 - \bar wB} \right)^{ - 1}$$ are Blaschke products for all |w| < 1 except possibly for w in a set of logarithmic capacity zero. If B is a Frostman Blaschke product, equivalently an inner multiplier for the space of Cauchy transforms of measures on the unit circle, we show that for all |w| < 1, B w is indeed another Frostman Blaschke product.

Ebenfelt, Peter; Viscardi, Michael

2006 Computational Methods and Function Theory

doi: 10.1007/BF03321636

We obtain and prove an explicit formula for the solution of the Dirichlet problem in a simply connected bounded domain in the plane with rational holomorphic data. In particular, the solution can be expressed using only two functions: the boundary data function and a Riemann map of the domain.

Lahiri, Indrajit; Yu, Kit-Wing

2006 Computational Methods and Function Theory

doi: 10.1007/BF03321637

Let p be a positive integer not less than 2. It is shown that a necessary condition for the generalized Fermat type functional equation $$\sum_{j=1}^pa_j(z){f_j}^{k_j}(z)\equiv 1$$ having non-constant meromorphic solutions f 1, f 2, …, f p is $$\sum_{j=1}^{p} {1\over k_{j}} \geq {1\over (p-1)+A_{p}}$$ , where A 2 = 1,2, A p = (2p − 3)/3 if p = 3, 4, 5, A p = (2p + 1 − 2√2p)/2 if p ≥ 6 and $$T\left( {r,a_j } \right) = \mathcal{O}\left( {T\left( {r,f_j } \right)} \right)$$ , 1 ≤ j ≤ p, as r → + ∞, r ∉ E and E is a set of finite linear measure. This improves the result of Yu and Yang [14] in 2002. Next we discuss a question of Hayman [7] and give a partial answer to it.