scholarly journals Ptolemy’s inequality, chordal metric, multiplicative metric

1982 ◽  
Vol 101 (2) ◽  
pp. 389-392 ◽  
Author(s):  
M. S. Klamkin ◽  
A. Meir
Keyword(s):  
Chordal Metric

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.

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]).

Ptolemy's Inequality and the Chordal Metric

1967 ◽  
Vol 40 (5) ◽  
pp. 233-235 ◽  
Author(s):  
Tom M. Apostol
Keyword(s):  
Chordal Metric

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.

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.

Ptolemy's Inequality and the Chordal Metric

1967 ◽  
Vol 40 (5) ◽  
pp. 233 ◽  
Author(s):  
Tom M. Apostol
Keyword(s):  
Chordal Metric
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