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On Hayman's alternative

  • J. K. Langley (a1)

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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.

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References

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1.Frank, G. and Mues, E.. Differentialpolynome (Oberwolfach, 1979).
2.Hayman, W. K.. Picard values of meromorphic functions and their derivatives. Ann. of Math., 70 (1959), 942.
3.Hayman, W. K.. Meromorphic Functions (Oxford, 1964).
4.Ince, E. L.. Ordinary Differential Equations (Dover, 1926).
5.Langley, J. K.. On differential polynomials and results of Hayman and Doeringer. Math. Zeit., 187 (1984), 111.
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