scholarly journals Some Results on Value Distribution of Meromorphic Functions in the Unit Disk

1969 ◽  
Vol 34 ◽  
pp. 105-119 ◽  
Author(s):  
Kam-Fook Tse
Keyword(s):  
Unit Circle ◽  
Unit Disk ◽  
Euclidean Distance ◽  
Meromorphic Functions ◽  
Riemann Sphere ◽  
Value Distribution ◽  
Open Unit ◽  
Open Unit Disk ◽  
Chordal Distance

Let C and D be the unit circle and the open unit disk respectively. We shall use p(z,z′) to represent the non-Euclidean distance [3, p. 263] between the two points z and z′ in D, and X(w, w′) to represent the chordal distance between the two points w and w′ on the Riemann Sphere Ω.

scholarly journals Intersections of Arc-Cluster Sets for Meromorphic Functions

1970 ◽  
Vol 40 ◽  
pp. 213-220 ◽  
Author(s):  
Charles L. Belna
Keyword(s):  
Unit Circle ◽  
Unit Disk ◽  
Complex Plane ◽  
Limit Point ◽  
Meromorphic Functions ◽  
Riemann Sphere ◽  
Open Unit ◽  
Open Unit Disk ◽  
Cluster Set ◽  
Jordan Arc

Let D and C denote the open unit disk and the unit circle in the complex plane, respectively; and let f be a function from D into the Riemann sphere Ω. An arc γ⊂D is said to be an arc at p∈C if γ∪{p} is a Jordan arc; and, for each t (0<t<1), the component of γ∩{z: t≤|z|<1} which has p as a limit point is said to be a terminal subarc of γ. If γ is an arc at p, the arc-cluster set C(f, p,γ) is the set of all points a∈Ω for which there exists a sequence {zk}a⊂γ with zk→p and f(zk)→a.

scholarly journals The three-arc and three-separated-arc properties of meromorphic functions

1974 ◽  
Vol 53 ◽  
pp. 137-140
Author(s):  
Frederick Bagemihl
Keyword(s):  
Meromorphic Function ◽  
Unit Circle ◽  
Unit Disk ◽  
Complex Plane ◽  
Meromorphic Functions ◽  
Riemann Sphere ◽  
Open Unit ◽  
Open Unit Disk

Let Γ be the unit circle and D be the open unit disk in the complex plane, and denote the Riemann sphere by Ω. Suppose that f(z) is a meromorphic function in D, and that ζ ∈ Γ.

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 The three-separated-arc property of the modular function

1976 ◽  
Vol 61 ◽  
pp. 203-204
Author(s):  
Frederick Bagemihl
Keyword(s):  
Unit Circle ◽  
Unit Disk ◽  
Complex Plane ◽  
Riemann Sphere ◽  
Modular Function ◽  
Open Unit ◽  
Open Unit Disk ◽  
Cluster Set

Let D be the open unit disk and Γ be the unit circle in the complex plane, and denote the Riemann sphere by Ω. If f(z) is a function defined on D with values belonging to Ω, if ζ ∈Γ, and if Λ is an arc at ζ then C∈(f, ζ) denotes the cluster set of f at ζ along Λ. If there exist three mutually exclusive arcs Λ1, Λ2, Λ3 at ζ such that

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.

scholarly journals Non-Essential Cluster Values and Normal Functions

1972 ◽  
Vol 47 ◽  
pp. 49-58
Author(s):  
C. L. Belna
Keyword(s):  
Unit Circle ◽  
Unit Disk ◽  
Riemann Sphere ◽  
Continuous Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Normal Functions ◽  
Cluster Values

We consider continuous functions f which map the open unit disk D into the Riemann sphere W. For a point ζ on the unit circle C, we say that χ is a chord at ζ if χ is a chord of C having one endpoint at ζ and that Δ is a Stolz angle at ζ if Δ is a Stolz angle with vertex ζ. Suppose S denotes either a chord at ζ, a Stolz angle at ζ, or D.

scholarly journals A Class of Nonlinear Integral Operators Preserving Double Subordinations

2008 ◽  
Vol 2008 ◽  
pp. 1-10 ◽  
Author(s):  
Oh Sang Kwon ◽  
Nak Eun Cho
Keyword(s):  
Unit Disk ◽  
Meromorphic Functions ◽  
Type Theorem ◽  
Integral Operators ◽  
Open Unit ◽  
Open Unit Disk ◽  
Sandwich Type ◽  
Space Of Meromorphic Functions ◽  
Nonlinear Integral Operators

The purpose of the present paper is to investigate some subordination- and superordination-preserving properties of certain integral operators defined on the space of meromorphic functions in the punctured open unit disk. The sandwich-type theorem for these integral operators is also considered.

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.

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.

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