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Sums and Products of Normal Functions

Published online by Cambridge University Press:  20 November 2018

David W. Bash
David W. Bash*
Affiliation:
Purdue University at Fort Wayne, Fort Wayne, Indiana
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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]).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 2 , April 1972 , pp. 261 - 269
Copyright
Copyright © Canadian Mathematical Society 1972

References

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