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On Hayman's alternative

Part of: Entire and meromorphic functions, and related topics

Published online by Cambridge University Press:  26 February 2010

J. K. Langley
J. K. Langley
Affiliation:
Department of Pure Mathematics, University of St. Andrews, St. Andrews, Fife, Scotland.
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Extract

Suppose that f(z) is non-constant and meromorphic in the plane and that, for some k≥= 1, a0(z),…, ak(z) are meromorphic in the plane with

for j' = 0,…, k. Here, using standard notation from [3], S(r,f) denotes any quantity satisfying S(r,f) = o(T(r,f)) as r→ ∞, possibly outside a set of finite linear measure. Then, setting

we have ([3, p. 57])

Theorem A. Suppose that f(z) is non-constant and meromorphic in the plane, and thatψ (z) given by (1.2) and (1.1) and is non-constant. Then

where N0(r, l/ψ') counts only zeros of ψ' which are not zeros of ψ − 1, and thecounting functions count points without regard to multiplicity.

MSC classification

Secondary: 30D35: Distribution of values, Nevanlinna theory
Type
Research Article
Information
Mathematika , Volume 32 , Issue 1 , June 1985 , pp. 139 - 146
Copyright
Copyright © University College London 1985

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References

1.Frank, G. and Mues, E.. Differentialpolynome (Oberwolfach, 1979).Google Scholar
2.Hayman, W. K.. Picard values of meromorphic functions and their derivatives. Ann. of Math., 70 (1959), 9–42.Google Scholar
3.Hayman, W. K.. Meromorphic Functions (Oxford, 1964).Google Scholar
4.Ince, E. L.. Ordinary Differential Equations (Dover, 1926).Google Scholar
5.Langley, J. K.. On differential polynomials and results of Hayman and Doeringer. Math. Zeit., 187 (1984), 1–11.CrossRefGoogle Scholar