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.