Computational Methods and Function Theory

Ostrovskii, Iossif

2013 Computational Methods and Function Theory

doi: 10.1007/BF03321069

Let P mn, 0 < Standardm < Standardn − 1, be a polynomial formed by the first m terms of the expansion of (1 + z)n according to the binomial formula. We show that, if m, n → ∞ in such a way that lim m,n →∞ m/n = α ∊ (0,1), then the zeros of P mn tend to a curve which can be explicitly described.

Chang, Jianming; Fang, Mingliang; Yang, Degui

2013 Computational Methods and Function Theory

doi: 10.1007/BF03321071

Let f be a non-constant meromorphic function satisfying f f′ ≠ 0, and let a α ≢ 0 be a small function related to f. If f(z) = a(z) whenever f′(z) = a(z), then either f ≡ f′ or f(z) = 2a/(1 − ce −2z ), where a, c are two non-zero constants and a(z) ≡ a.

Aleksandrov, Alexei

2013 Computational Methods and Function Theory

doi: 10.1007/BF03321072

A function u on the unit circle T is said to be badly approximable in the weighted space L p(T,w)} if ¦¦u + f¦¦ L p(T,w) ≥¦L p(T,w) for all f ∊ H∞. We prove that if an unimodular function u is badly approximable in L p(T, w}) for all p ∊ (0, +∞) and some non-zero weight w, then $\overline{u}$ is an inner function. We describe the inner functions Θ and the weights w on the unit circle T such that Θ is badly approximable in L p(T, w) for all p > 0. It turns out that, for given inner functions Θ, the class of all weights satisfying the above-mentioned condition depends only on the zero set of Θ. In other words, Θ is badly approximable in L p(T, w) for all p ∊ (0, +∞) if and only if $\overline{B}$ is badly approximable in L p(T,w) for all p ∊ (0, +∞), where B is a Blaschke product with simple zeros and such that Θ−1(0) = B−1(0).

Clifford, Eleanor

2013 Computational Methods and Function Theory

doi: 10.1007/BF03321073

We consider the differential operator Λk defined by $$\Lambda_k(y) = \Psi_k(y) + a_{k-1}\Psi_{k-1}(y) +... + a_1\Psi_1(y) +a_0,$$ where a 0,…, a k−1 are analytic functions of restricted growth and Ψk(y) is a differential operator defined by Ψ1(y) = y and Ψk+1(y) = yΨk(y) + (Ψk(y))′ for k ∊ N. We suppose that k ≥ 3, that F is a meromorphic function on an annulus A(r 0), and that Λk(F) has all its zeros on a set E such that E has no limit point in A(r 0). We suppose also that all simple poles a of F in A(r0) E have res(F, a) ∉ {1,…,k − 1}. We then deduce that F is a function of restricted growth in the Nevanlinna sense. This extends a theorem of Bergweiler and Langley [1]. We show also that this result does not hold when a0,…, a k−1 are meromorphic functions.

Üreyen, Adem

2013 Computational Methods and Function Theory

doi: 10.1007/BF03321074

A maximum modulus point of an entire function f is a point w such that ¦f(w)¦ = max¦f(z)¦: ¦z¦ = ¦w¦. Denote by R(w,f) the distance between a maximum modulus point w and the zero set of f. In 1938, A. J. Macintyre obtained lower asymptotic estimates for R(w, f) as ¦w¦ →∞ valid outside of an exceptional set. The problem of asymptotic estimates valid for all sufficiently large ¦w¦ was studied by I. V. Ostrovskii and the author for functions of finite positive order. In this paper, we study this problem for functions of either zero or infinite order.

Broch, Ole

2013 Computational Methods and Function Theory

doi: 10.1007/BF03321078

Relations involving ordered quadruples of points on the boundary of a Jordan domain are investigated, and it is shown that these relations hold if and only if the domain is a John disk. In particular connections between absolute cross ratios and moduli of corresponding quadrilaterals are examined. A characterization of John disks based on a relation between diameters of subarcs of the boundary and bounds for moduli of associated quadrilaterals is also given. The results are one sided analogues to characterizations of quasidisks.

Bshouty, Daoud; Lyzzaik, Abdallah

2013 Computational Methods and Function Theory

doi: 10.1007/BF03321079

A necessary and sufficient condition is given for a planar harmonic mapping f to be locally decomposable as a univalent harmonic mapping F of an analytic function φ.

Akeroyd, John; Karber, Kristi

2013 Computational Methods and Function Theory

doi: 10.1007/BF03321081

Our work addresses the question: for which (infinite) Blaschke products B does the star-shift invariant subspace K b:= H 2(D) Θ BH 2(D) contain a (non-trivial) function with a non-trivial singular inner factor? In the case that the zeros of B have only finitely many accumulation points w 1,w 2, …, w n in T, a recent paper shows that, for an affirmative answer, there necessarily exist k, 1 ≤ k ≤ n, and a subsequence of the zeros of B that converges tangentially to w k on “both sides” of w k. One of the results in this article improves upon this theorem. And, currently, the only examples of Blaschke products in the literature that are shown to yield an affirmative answer are those that have a proper factor b that satisfies b(D) ≠ D. We produce many examples here that have no such factor.