Radial and Angular Limits of Meromorphic Functions

1963 ◽  
Vol 15 ◽  
pp. 471-474 ◽  
Author(s):  
G. T. Cargo
Keyword(s):  
Meromorphic Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Classical Theorem ◽  
Radial Limit ◽  
Angular Limit ◽  
Normal Functions ◽  
Bounded Holomorphic Function ◽  
Bounded Holomorphic ◽  
Angular Limits

Let us say that a function denned in the open unit disk D has the Montel property if the set of those points eiθ on the unit circle C where the radial limit exists coincides with the set where the angular limit exists. By a classical theorem of Montel (4), every bounded holomorphic function has this property. Meromorphic functions omitting at least three values and, more generally, the normal functions recently introduced by Lehto and Virtanen (3) also enjoy the Montel property (also see 1).

scholarly journals On Angular Limits of Normal Meromorphic Functions: A Geometric Aspect

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Zarko Pavicevic
Keyword(s):  
Meromorphic Functions ◽  
Sufficient Conditions ◽  
Open Unit ◽  
Geometric Aspect ◽  
Limit Values ◽  
Angular Limit ◽  
Normal Meromorphic Function ◽  
Lindelöf Theorem ◽  
Necessary And Sufficient ◽  
Angular Limits

We will prove the assertions which give necessary and sufficient conditions for a normal meromorphic function on the open unit disk to have an angular limit. The results obtained show that the conditions from the classical Lindelöf theorem, as well as the theorems of Lehto and Virtanen and Bagemihl and Seidel, concerning angular limit values of meromorphic functions, can be weakened.

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.

scholarly journals On the Relation between a Cluster set Introduced by Constantinescu and Cornea, and the Fine Cluster set of Cartan, Brelot, and Naïm

1965 ◽  
Vol 8 (1) ◽  
pp. 59-71
Author(s):  
H. L. Jackson
Keyword(s):  
Analytic Function ◽  
Holomorphic Function ◽  
Limit Theorems ◽  
Function Theory ◽  
Angular Limit ◽  
Boundary Limit ◽  
Cluster Set ◽  
Almost Everywhere ◽  
Bounded Holomorphic Function ◽  
Bounded Holomorphic

The field of boundary limit theorems in analytic function theory is usually considered to have begun about 1906, with the publication of Fatou's thesis [8]. In this remarkable memoir a theorem is proved, that now bears the author's name, which implies that any bounded holomorphic function defined on the unit disk possesses an angular limit almost everywhere (Lebesgue measure) on the frontier. Outstanding classical contributions to this field can be attributed to F. and M. Riesz, R. Nevanlinna, Lusin, Privaloff, Frostman, Plessner, and others.

A Filter Description of the Homomorphisms of H∞

1965 ◽  
Vol 17 ◽  
pp. 734-757 ◽  
Author(s):  
A. Kerr-Lawson
Keyword(s):  
Unit Disk ◽  
Maximal Ideal ◽  
Continuous Mapping ◽  
Open Unit ◽  
Supremum Norm ◽  
Ideal Space ◽  
Open Unit Disk ◽  
Classical Theorem ◽  
Bounded Analytic Functions ◽  
The Hyperbolic Metric

The algebra of bounded analytic functions on the open unit disk D, usually written H∞ i is a commutative Banach algebra under the supremum norm. Since its compact maximal ideal space M (space of complex homomorphisms) is an extension space of the unit disk, there must be a continuous mapping form βD, the Stone-Čech compactification of D, onto M. R. C. Buck has remarked (4), that this mapping fails to be one-one, in the light of a classical theorem of Pick. If the points of βD are represented by filters of subsets of D, we can identify those filters which are sufficiently close in terms of the hyperbolic metric on D in an attempt to get a one-one correspondence between filters and points of M.

scholarly journals On Metric Properties of Sets of Angular Limits of Meromorphic Functions

1966 ◽  
Vol 26 ◽  
pp. 121-126 ◽  
Author(s):  
J. E. Mcmillan
Keyword(s):  
Harmonic Measure ◽  
Unit Disc ◽  
Meromorphic Functions ◽  
Linear Measure ◽  
Closed Set ◽  
Angular Limit ◽  
Metric Properties ◽  
Angular Limits

Let f be a nonconstant function meromorphic in the unit disc , with circumference C, and let Ez be a subset of C with positive (linear) measure. Suppose that at each ζ ∈ Ezf has an angular limit aζ and let It is known that Ew contains a closed set with positive harmonic measure (see Priwalow [6, p. 210] or Tsuji [7, p. 339]).

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

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 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 Starlikeness Properties of a New Integral Operator for Meromorphic Functions

2011 ◽  
Vol 2011 ◽  
pp. 1-8 ◽  
Author(s):  
Aabed Mohammed ◽  
Maslina Darus
Keyword(s):  
Integral Operator ◽  
Unit Disk ◽  
Meromorphic Functions ◽  
Open Unit ◽  
Open Unit Disk

We define here an integral operator for meromorphic functions in the punctured open unit disk. Several starlikeness conditions for the integral operator are derived.

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.

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