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IDENTITY THEOREMS FOR FUNCTIONS OF BOUNDED CHARACTERISTIC

Published online by Cambridge University Press:  01 August 1998

W. K. HAYMAN
W. K. HAYMAN
Affiliation:
Department of Mathematics, Imperial College, 180 Queen's Gate, London SW7 2BZ
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Abstract

Suppose that f(z) is a meromorphic function of bounded characteristic in the unit disk Δ[ratio ][mid ]z[mid ]<1. Then we shall say that f(z)∈N. It follows (for example from [3, Lemma 6.7, p. 174 and the following]) that

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where h1(z), h2(z) are holomorphic in Δ and have positive real part there, while Π1(z), Π2(z) are Blaschke products, that is,

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where p is a positive integer or zero, 0<[mid ]aj[mid ]<1, c is a constant and

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We note in particular that, if c≠0, so that f(z)[nequiv ]0,

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so that f(z)=0 only at the points aj. Suppose now that zj is a sequence of distinct points in Δ such that

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If f(zj)=0 for each j and f∈N, then f(z)≡0.

N. Danikas [1] has shown that the same conclusion obtains if f(zj)→0 sufficiently rapidly as j→∞. Let εj, λj be sequences of positive numbers such that

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Danikas then defines

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and proves Theorem A.

Type
Notes and Papers
Information
Journal of the London Mathematical Society , Volume 58 , Issue 1 , August 1998 , pp. 127 - 140
Copyright
The London Mathematical Society 1998

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