scholarly journals Meromorphic Functions with Large Sets of Julia Points

1973 ◽  
Vol 50 ◽  
pp. 1-6
Author(s):  
Peter Colwell
Keyword(s):  
Meromorphic Functions ◽  
Riemann Sphere ◽  
Large Sets ◽  
Chordal Metric

Let D = {z : |z| < 1} and C = {z : |z| = 1}. If W denotes the Riemann sphere equipped the chordal metric X, let f: D → W be meromorphic. A chord T lying in D except for an endpoint γ ∈ C is called a Julia segment for f if for each Stolz angle Δ in D at γ which contains T, f assumes infinitely often in Δ all values of W with at most two exceptions. We call γ ∈ C a Julia point for f if every chord in D ending at γ is a Julia segment for f, and we denote by J(f) the set of Julia points of f.

Possibility of Uniform Rational Approximation in the Spherical Metric

1976 ◽  
Vol 28 (1) ◽  
pp. 112-115 ◽  
Author(s):  
P. M. Gauthier ◽  
A. Roth ◽  
J. L. Walsh
Keyword(s):  
Compact Subset ◽  
Rational Approximation ◽  
Complex Plane ◽  
Riemann Sphere ◽  
Rational Functions ◽  
Finite Complex ◽  
Finite Plane ◽  
Extended Plane ◽  
Chordal Metric

Let ƒ b e a mapping defined on a compact subset K of the finite complex plane C and taking its values on the extended plane C ⋃ ﹛ ∞﹜. For x a metric on the extended plane, we consider the possibility of approximating ƒ x-uniformly on K by rational functions. Since all metrics on C ⋃ ﹛oo ﹜ are equivalent, we shall consider that x is the chordal metric on the Riemann sphere of diameter one resting on a finite plane at the origin.

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.

Sums and Products of Normal Functions

1972 ◽  
Vol 24 (2) ◽  
pp. 261-269
Author(s):  
David W. Bash
Keyword(s):  
Riemann Sphere ◽  
Normal Function ◽  
Hyperbolic Distance ◽  
Normal Functions ◽  
Euclidean Metric ◽  
A Value ◽  
Chordal Metric ◽  
Bounded Characteristic ◽  
Definition Of ◽  
Complex Valued

Let D be the unit disk in the complex plane. Let p(z, z’) denote the hyperbolic distance between z and z’ in ((1 + u)/ (1 — u)) = tanh-1 u, [6, chapter 15]). Let W be the Riemann sphere with the chordal metric. A complex valued function F(Z) in D is a normal function if lor each pair of sequences {zn} and {zn’} of points in D such that the convergence of {(fzn)} to a value α in W implies the convergence of {f(zn’)} to α. Two sequences {zn} and {zn’} of points in D are called close sequences if ρ(zn, zn’) → 0. (There are several equivalent definitions of normality if the functions are meromorphic.) The definition of a normal function implies that a normal function is continuous at each point of D when using the Euclidean metric in the domain and the chordal metric in the range.We wish to study the sums and products of normal functions. Some functions, such as a function in a Hardy p-class, p > 0, (or really any function of bounded characteristic) can be written as a sum or product of two normal functions, but sums and products of normal functions need not be normal (see Lappan [7]).

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.

Cercles De Remplissage and Asymptotic Behaviour along Circuitous Paths

1970 ◽  
Vol 22 (2) ◽  
pp. 389-393 ◽  
Author(s):  
P. M. Gauthier
Keyword(s):  
Meromorphic Function ◽  
Asymptotic Behaviour ◽  
Metric Space ◽  
Unit Disc ◽  
Riemann Sphere ◽  
Holomorphic Functions ◽  
Weak Form ◽  
Value Distribution ◽  
Hyperbolic Metric ◽  
Chordal Metric

In this paper we consider the value distribution of a meromorphic function whose behaviour is prescribed along a spiral. The existence of extremely wild holomorphic functions is established. Indeed a very weak form of one of our results would be that there are holomorphic functions (in the unit disc or the plane) for which every curve “tending to the boundary” is a Julia curve.The theorems in this paper generalize results of Gavrilov [7], Lange [9], and Seidel [11].I wish to express my thanks to Professor W. Seidel for his guidance and encouragement.2. Preliminaries. For the most part we will be dealing with the metric space (D, ρ) where D is the unit disc, |z| < 1, and ρ is the non-Euclidean hyperbolic metric on D. The chordal metric on the Riemann sphere will be denoted by x.

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 Normal Light Interior Functions Defined in the Unit Disk

1970 ◽  
Vol 39 ◽  
pp. 149-155 ◽  
Author(s):  
J.H. Mathews
Keyword(s):  
Continuous Function ◽  
Unit Circle ◽  
Unit Disk ◽  
Riemann Sphere ◽  
Hyperbolic Metric ◽  
Chordal Metric ◽  
Normal Light ◽  
Uniformly Continuous

Let D be the unit disk, C the unit circle, and f a continuous function from D into the Riemann sphere W. We say that f is normal if f is uniformly continuous with respect to the non-Euclidean hyperbolic metric in D and the chordal metric in W.

Further investigations on Fujimoto type strong uniqueness polynomials

Filomat ◽  
2017 ◽  
Vol 31 (16) ◽  
pp. 5203-5216
Author(s):  
Abhijit Banerjee ◽  
Bikash Chakraborty ◽  
Sanjay Mallick
Keyword(s):  
Meromorphic Functions ◽  
Future Research ◽  
Strong Uniqueness ◽  
Open Questions

Taking the question posed by the first author in [1] into background, we further exhaust-ably investigate existing Fujimoto type Strong Uniqueness Polynomial for Meromorphic functions (SUPM). We also introduce a new kind of SUPM named Restricted SUPM and exhibit some results which will give us a new direction to discuss the characteristics of a SUPM. Moreover, throughout the paper, we pose a number of open questions for future research.

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