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The Maximum Modulus of Normal Meromorphic Functions and Applications to Value Distribution

Published online by Cambridge University Press:  20 November 2018

Paul Gauthier
Paul Gauthier*
Affiliation:
Université de Montréal, Montréal, Québec
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Let f(z) be a function meromorphic in the unit disc D = (|z| < 1). We consider the maximum modulus

and the minimum modulus

When no confusion is likely, we shall write M(r) and m(r) in place of M(r,f) and m(r,f).

Since every normal holomorphic function belongs to an invariant normal family, a theorem of Hayman [6, Theorem 6.8] yields the following result.

THEOREM 1. If f(z) is a normal holomorphic function in the unit disc D, then

(1)

This means that for normal holomorphic functions, M(r) cannot grow too rapidly. The main result of this paper (Theorem 5, also due to Hayman, but unpublished) is that a similar situation holds for normal meromorphic functions.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 4 , 01 August 1970 , pp. 803 - 814
Copyright
Copyright © Canadian Mathematical Society 1970

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