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 On a Uniqueness Theorem

1969 ◽  
Vol 35 ◽  
pp. 151-157 ◽  
Author(s):  
V. I. Gavrilov
Keyword(s):  
Unit Circle ◽  
Uniqueness Theorem ◽  
Unit Disk ◽  
Complex Plane ◽  
Open Unit ◽  
Open Unit Disk ◽  
Extended Complex ◽  
Extended Complex Plane

1. Let D be the open unit disk and r be the unit circle in the complex plane, and denote by Q the extended complex plane or the Rie-mann sphere.

scholarly journals A Property of the Principal Cluster sets of a Class of Holomorphic Functions

1971 ◽  
Vol 43 ◽  
pp. 157-159
Author(s):  
F. Bagemihl
Keyword(s):  
Meromorphic Function ◽  
Unit Circle ◽  
Unit Disk ◽  
Complex Plane ◽  
Riemann Sphere ◽  
Holomorphic Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Cluster Set ◽  
Principal Cluster

Let D be the open unit disk and Γ be the unit circle in the complex plane, and denote by Ω the Riemann sphere. If f(z) is a meromorphic function in D, and if ζ∈Г, then the principal cluster set of f at ζ is the set

scholarly journals A Note on Extended Ambiguous Points

1971 ◽  
Vol 43 ◽  
pp. 167-168
Author(s):  
J.L. Stebbins
Keyword(s):  
Unit Circle ◽  
Unit Disk ◽  
Complex Plane ◽  
Arbitrary Function ◽  
Riemann Sphere ◽  
Open Unit ◽  
Open Unit Disk ◽  
Ambiguous Point

Let f be an arbitrary function from the open unit disk D of the complex plane into the Riemann sphere S. If p is any point on the unit circle C, C(f, p) is the set of all points w such that there exists in D a sequence of points {Zj} such that zj→p and f(zj)→w. CΔ(f, p) is defined in the same way, but the sequence {Zj} is restricted to Δ⊂D. If α and β are two arcs in D terminating at p and Cα(f, p)∩Cβ(f, p) = Φ, p is called an ambiguous point for f.

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).

Multipliers of Fractional Cauchy Transforms and Smoothness Conditions

1998 ◽  
Vol 50 (3) ◽  
pp. 595-604 ◽  
Author(s):  
Donghan Luo ◽  
Thomas Macgregor
Keyword(s):  
Analytic Function ◽  
Unit Circle ◽  
Unit Disk ◽  
Borel Measure ◽  
Open Unit ◽  
Open Unit Disk ◽  
Cauchy Transforms ◽  
The Family ◽  
Complex Borel Measure

AbstractThis paper studies conditions on an analytic function that imply it belongs to Mα, the set of multipliers of the family of functions given by where μ is a complex Borel measure on the unit circle and α > 0. There are two main theorems. The first asserts that if 0 < α < 1 and sup. The second asserts that if 0 < α < 1, ƒ ∈ H∞ and supt. The conditions in these theorems are shown to relate to a number of smoothness conditions on the unit circle for a function analytic in the open unit disk and continuous in its closure.

The Maximal Ideal Space of H∞ + C on the ball in Cn

1979 ◽  
Vol 31 (1) ◽  
pp. 79-86 ◽  
Author(s):  
Gerard Mcdonald
Keyword(s):  
Maximal Ideal ◽  
Banach Algebras ◽  
Holomorphic Functions ◽  
Maximal Ideal Space ◽  
Open Unit ◽  
Ideal Space ◽  
Open Unit Ball ◽  
Closed Subalgebra ◽  
Image Position ◽  
Bounded Holomorphic

Let S denote the unit sphere in Cn, B the (open) unit ball in Cn and H∞(B) the collection of all bounded holomorphic functions on B. For f ∈ H∞(B) the limitsexist for almost every ζ in S, and the map ƒ → ƒ* defines an isometric isomorphism from H∞(B) onto a closed subalgebra of L∞(S), denoted H∞(S). (The only measure on S we will refer to in this paper is the Lebesgue measure, dσ, generated by Euclidean surface area.) Rudin has shown in [4] that the spaces H∞(B) + C(B) and H∞(S) + C(S) are Banach algebras in the sup norm. In this paper we will show that the maximal ideal space of H∞(B) + C(B), Σ (H∞(B) + C(B)), is naturally homeomorphic to Σ (H∞(B)) and that Z (H∞(S) + C(S)) is naturally homeomorphic to Σ (H∞(S))\B.

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 Some Results and Problems Concerning Chordal Principal Cluster Sets

1967 ◽  
Vol 29 ◽  
pp. 7-18 ◽  
Author(s):  
F. Bagemihl
Keyword(s):  
Meromorphic Function ◽  
Unit Circle ◽  
Unit Disk ◽  
Riemann Sphere ◽  
Open Unit ◽  
Open Unit Disk ◽  
Continuous Curve ◽  
Cluster Set ◽  
Asymptotic Values ◽  
Principal Cluster

Let Γ be the unit circle and D be the open unit disk in the complex plane, and denote the Riemann sphere by Ω. By an arc at a point ζ∈Γ we mean a continuous curve such that |z(t)| < 1 for 0 ≦ t < 1 and . A terminal subarc of an arc Λ at ζ is a subarc of the form z = z (t) (t0 ≦ t < 1), where 0 ≦ t0<1. Suppose that f(z) is a meromorphic function in D. Then A(f) denotes the set of asymptotic values of f; and if ζ∈Γ, then C(f, ζ) means the cluster set of f at ζ and is the outer angular cluster set of f at ζ (see [13]).

scholarly journals On a problem of Bonar concerning Fatou points for annular functions

1975 ◽  
Vol 56 ◽  
pp. 163-170
Author(s):  
Akio Osada
Keyword(s):  
Unit Circle ◽  
Unit Disk ◽  
Open Unit ◽  
Open Unit Disk ◽  
Minimum Modulus ◽  
Jordan Curves ◽  
Fatou Points ◽  
Annular Functions

The purpose of this paper is to study the distribution of Fatou points of annular functions introduced by Bagemihl and Erdös [1]. Recall that a function f(z), regular in the open unit disk D: | z | < 1, is referred to as an annular function if there exists a sequence {Jn} of closed Jordan curves, converging out to the unit circle C: | z | = 1, such that the minimum modulus of f(z) on Jn increases to infinity. If the Jn can be taken as circles concentric with C, f(z) will be called strongly annular.

scholarly journals Continuations of Analytic Functions of Class S and Class U

1970 ◽  
Vol 40 ◽  
pp. 33-37
Author(s):  
Shinji Yamashita
Keyword(s):  
Unit Circle ◽  
Unit Disk ◽  
Analytic Functions ◽  
Inner Function ◽  
Open Unit ◽  
Open Unit Disk ◽  
Radial Limit

Let f be of class U in Seidel’s sense ([4, p. 32], = “inner function” in [3, p. 62]) in the open unit disk D. Then f has, by definition, the radial limit f(eiθ) of modulus one a.e. on the unit circle K. As a consequence of Smirnov’s theorem [5, p. 64] we know that the function

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