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A note on a problem of Gauthier
Published online by Cambridge University Press: 26 February 2010
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Let D denote the unit disc in the complex plane. If p and q are two complex numbers, let x(p, q) denote the chordal distance between p and q on the Riemann sphere. In particular, we have the formula
and
The following problem was posed by Paul Gauthier: if f(z) and g(z) are meromorphic functions in D such that Clunie [4] has answered this problem in the negative by constructing different meromorphic functions f(z) and g(z) with the desired property. However, the functions constructed by Clunie both have an infinity of poles in D. It is the purpose of this note to give an example of two analytic functions―which thus have no poles―which also give a negative answer to the problem of Gauthier.
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- Copyright © University College London 1971
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