Zero Tracts of Blaschke Products

1966 ◽  
Vol 18 ◽  
pp. 1072-1078 ◽  
Author(s):  
C. N. Linden ◽  
H. Somadasa
Keyword(s):  
Unit Disk ◽  
Blaschke Product ◽  
Open Unit ◽  
Complex Numbers ◽  
Open Unit Disk ◽  
Blaschke Products ◽  
Image Position

Let ﹛an﹜ be a sequence of complex numbers such thatandThen {an} is called a Blaschke sequence. For each Blaschke sequence {an} a Blaschke product is defined asThus a Blaschke product B(z, ﹛an﹜) is a function regular in the open unit disk D = {z: |z| < 1﹜ and having a zero at each point of the sequence ﹛an﹜.

Approximation by Unimodular Functions

1971 ◽  
Vol 23 (2) ◽  
pp. 257-269 ◽  
Author(s):  
Stephen Fisher
Keyword(s):  
Blaschke Product ◽  
Uniform Approximation ◽  
Unit Disc ◽  
Open Unit ◽  
Blaschke Products ◽  
Open Unit Disc ◽  
Pointwise Approximation ◽  
Finite Blaschke Product ◽  
Image Position ◽  
Restricted Zeros

The theorems in this paper are all concerned with either pointwise or uniform approximation by functions which have unit modulus or by convex combinations of such functions. The results are related to, and are outgrowths of, the theorems in [4; 5; 10].In § 1, we show that a function bounded by 1, which is analytic in the open unit disc Δ and continuous on may be approximated uniformly on the set where it has modulus 1 (subject to certain restrictions; see Theorem 1) by a finite Blaschke product; that is, by a function of the form*where |λ| = 1 and |αi| < 1, i = 1, …, N. In § 1 we also discuss pointwise approximation by Blaschke products with restricted zeros.

Behavior of Coefficients of Inverses of α-Spiral Functions

1986 ◽  
Vol 38 (6) ◽  
pp. 1329-1337 ◽  
Author(s):  
Richard J. Libera ◽  
Eligiusz J. Złotkiewicz
Keyword(s):  
Unit Disk ◽  
Series Expansion ◽  
Open Unit ◽  
Open Unit Disk ◽  
Maclaurin Series ◽  
Koebe Function ◽  
One To One ◽  
Image Position

If f(z) is univalent (regular and one-to-one) in the open unit disk Δ, Δ = {z ∊ C:│z│ < 1}, and has a Maclaurin series expansion of the form(1.1)then, as de Branges has shown, │ak│ = k, for k = 2, 3, … and the Koebe function.(1.1)serves to show that these bounds are the best ones possible (see [3]). The functions defined above are generally said to constitute the class .

scholarly journals Recursive interpolating sequences

2018 ◽  
Vol 16 (1) ◽  
pp. 461-468
Author(s):  
Francesc Tugores
Keyword(s):  
Analytic Function ◽  
Unit Disk ◽  
Blaschke Product ◽  
Bounded Sequence ◽  
Open Unit ◽  
Open Unit Disk ◽  
Separation Condition ◽  
Interpolating Sequences ◽  
Interpolation Problems ◽  
Suitable Sequence

AbstractThis paper is devoted to pose several interpolation problems on the open unit disk 𝔻 of the complex plane in a recursive and linear way. We look for interpolating sequences (zn) in 𝔻 so that given a bounded sequence (an) and a suitable sequence (wn), there is a bounded analytic function f on 𝔻 such that f(z1) = w1 and f(zn+1) = anf(zn) + wn+1. We add a recursion for the derivative of the type: f′(z1) = $\begin{array}{} w_1' \end{array} $ and f′(zn+1) = $\begin{array}{} a_n' \end{array} $ [(1 − |zn|2)/(1 − |zn+1|2)] f′(zn) + $\begin{array}{} w_{n+1}', \end{array} $ where ($\begin{array}{} a_n' \end{array} $) is bounded and ($\begin{array}{} w_n' \end{array} $) is an appropriate sequence, and we also look for zero-sequences verifying the recursion for f′. The conditions on these interpolating sequences involve the Blaschke product with zeros at their points, one of them being the uniform separation condition.

scholarly journals Disk-like functions

1970 ◽  
Vol 11 (2) ◽  
pp. 251-256
Author(s):  
Richard J. Libera
Keyword(s):  
Unit Disk ◽  
Geometric Property ◽  
Starlike Functions ◽  
Positive Real Part ◽  
Open Unit ◽  
Open Unit Disk ◽  
Positive Real ◽  
Image Position

The class s of functions f(z) which are regular and univalent in the open unit disk △ = {z: |z| < 1} each normalized by the conditionshas been studied intensively for over fifty years. A large and very successful portion of this work has dealt with subclasses of L characterized by some geometric property of f[Δ], the image of Δ under f(z), which is expressible in analytic terms. The class of starlike functions in L is one of these [3]; f(z) is starlike with respect to the origin if the segment [0,f(z)] is in f[Δ] for every z in Δ and this condition is equivalent to requiring that have a positive real part in Δ.

On the Growth of Blaschke Products

1969 ◽  
Vol 21 ◽  
pp. 595-601 ◽  
Author(s):  
G. R. MacLane ◽  
L. A. Rubel
Keyword(s):  
Angular Distribution ◽  
Analytic Function ◽  
Blaschke Product ◽  
Bounded Function ◽  
Related Question ◽  
Complex Numbers ◽  
Private Communication ◽  
Blaschke Products ◽  
The Subject ◽  
Image Position

It is well known that the distribution of the zeros of an analytic function affects its rate of growth. The literature is too extensive to indicate here. We only point out (1, p. 27; 2; 3; 5), where the angular distribution of the zeros plays a role, as it will in this paper. In private communication, A. Zygmund has raised the following related question, which is the subject of our investigation here.Let {zn}, n = 1, 2, 3, …, be a sequence of non-zero complex numbers of modulus less than 1, such that ∑(1 – |zn|) < ∞, and consider the Blaschke product1Let2What are the sequences {zn} for which I(r) is a bounded function of r?

Almost-Bounded Holomorphic Functions with Prescribed Ambiguous Points

1964 ◽  
Vol 16 ◽  
pp. 231-240 ◽  
Author(s):  
G. T. Cargo
Keyword(s):  
Unit Circle ◽  
Unit Disk ◽  
Holomorphic Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Extended Complex ◽  
Image Position ◽  
Bounded Holomorphic ◽  
Jordan Arcs ◽  
Ambiguous Point

Let f be a function mapping the open unit disk D into the extended complex plane. A point ζ on the unit circle C is called an ambiguous point of f if there exist two Jordan arcs J1 and J2, each having an endpoint at ζ and lying, except for ζ, in D, such that

scholarly journals Multipliers of Bergman spaces into Lebesgue spaces

1986 ◽  
Vol 29 (1) ◽  
pp. 125-131 ◽  
Author(s):  
Daniel H. Luecking
Keyword(s):  
Lebesgue Measure ◽  
Unit Disk ◽  
Analytic Functions ◽  
Lebesgue Space ◽  
Borel Measure ◽  
Bergman Spaces ◽  
Open Unit ◽  
Open Unit Disk ◽  
Lebesgue Spaces ◽  
Image Position

Let U be the open unit disk in the complex plane endowed with normalized Lebesgue measure m. will denote the usual Lebesgue space with respect to m, with 0<p<+∞. The Bergman space consisting of the analytic functions in will be denoted . Let μ be some positivefinite Borel measure on U. It has been known for some time (see [6] and [9]) what conditions on μ are equivalent to the estimate: There is a constant C such thatprovided 0<p≦q.

On Algebras Generated by Composition Operators

1974 ◽  
Vol 26 (5) ◽  
pp. 1234-1241 ◽  
Author(s):  
J. A. Cima ◽  
W. R. Wogen
Keyword(s):  
Banach Spaces ◽  
Unit Disk ◽  
Complex Plane ◽  
Hardy Spaces ◽  
Composition Operators ◽  
Open Unit ◽  
Open Unit Disk ◽  
Group Of Automorphisms ◽  
Image Position

Let Δ be the open unit disk in the complex plane and let be the group of automorphisms of Δ onto Δ, define byThe Banach spaces Hp = Hp(Δ), 1 ≦ p < ∞, are the Hardy spaces of functions analytic in Δ with their integral p means bounded,

Green's Potentials with Prescribed Boundary Values

1977 ◽  
Vol 29 (4) ◽  
pp. 681-686
Author(s):  
Jang-Mei G. Wu
Keyword(s):  
Green’S Function ◽  
Unit Disk ◽  
Mass Distribution ◽  
Green's Function ◽  
Open Unit ◽  
Open Unit Disk ◽  
Boundary Values ◽  
Image Position

Let U, C denote the open unit disk and unit circumference, respectively and G(z, w) be the Green's function on U. We say v is the Green's potential of a mass distribution v on U if

Normal Functions and Non-Tangential Boundary Arcs

1966 ◽  
Vol 18 ◽  
pp. 256-264 ◽  
Author(s):  
P. Lappan ◽  
D. C. Rung
Keyword(s):  
Unit Circle ◽  
Unit Disk ◽  
Complex Plane ◽  
Hyperbolic Geometry ◽  
Open Unit ◽  
Open Unit Disk ◽  
Normal Functions ◽  
Image Position

Let D and C denote respectively the open unit disk and the unit circle in the complex plane. Further, γ = z(t), 0 ⩽ t ⩽ 1, will denote a simple continuous arc lying in D except for Ƭ = z(l) ∈ C, and we shall say that γ is a boundary arc at Ƭ.We use extensively the notions of non-Euclidean hyperbolic geometry in D and employ the usual metricwhere a and b are elements of D. For a ∈ D and r > 0 letFor details we refer the reader to (4).

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