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.