Identity Theorems for Functions of Bounded Characteristic |
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Authors: | Hayman W K |
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Institution: | 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 characteristicin the unit disk :|z|<1. Then we shall say that f(z)N. Itfollows (for example from 3, Lemma 6.7, p. 174 and the following])that
where h1(z), h2(z) are holomorphic in and have positive realpart there, while 1(z), 2(z) are Blaschke products, that is,
where p is a positive integer or zero, 0<|aj|<1, c isa constant and (1|aj|)<. We note in particular that, if c0, so that f(z)0,
(1.1) so that f(z)=0 only at the points aj. Suppose now that zj isa sequence of distinct points in such that |zj|1 as j and (1|zj|)=. (1.2) If f(zj)=0 for each j and fN, then f(z)0. N. Danikas 1] has shown that the same conclusion obtains iff(zj)0 sufficiently rapidly as j. Let j, j be sequences of positivenumbers such that j< and j as j. Danikas then defines
and proves Theorem A. |
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