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Identity theorems for functions of bounded characteristic. (English) Zbl 0922.30005
Suppose that \(f(z)\) is meromorphic of bounded characteristic in the unit disk \(\Delta: | z|< 1\). Then we say that \(f(z)\in {\mathcal N}\). It is classical in this case, that if \(z_j\in \Delta\), \[ | z_j |\to 1 \quad \text{as } j\to \infty, \qquad \text{and} \quad \sum(1- | z_j|)= \infty, \tag{1} \] then \(f(z_j)=0\) for all \(j\) implies \(f(z) \equiv 0\).
N. Danikas [J. Approximation Theory 77, No. 2, 184-190 (1994; Zbl 0804.30031)] has shown that it is enough for this conclusion that \(f(z_j) \to 0\) sufficiently rapidly, depending on the sequence \(z_j\).
In this paper a sharper form of Danikas’ theorem is proved. We define \[ \eta_j= \prod_k \Biggl\{ {{2| z_j- z_k|} \over {1-| z_j |}} \Biggr\}. \] Here the product is taken over all \(k\) such that \(0<| z_k- z_j|< {1\over 2}(1- | z_j|)\).
Theorem 2. If \(f(z)\in {\mathcal N}\) and (1) holds, then either \(f(z) \equiv 0\), or, for a sequence \(j= j_p\), we have \[ \log | f(z_j) |\geq \log \eta_j+ {{O(1)} \over {1-| z_j|}}. \] Examples are given to show that Theorem 2 is “nearly” sharp, at least for a sequence \(\{z_j\}\) lying in a Stolz angle.
At the other end of the scale we may ask for what sequences \(\{z_j \}\) merely \[ f(z_j)\to 0 \quad \text{implies } f(z) \equiv 0, \qquad \text{whenever } f\in {\mathcal N}. \tag{2} \] Theorem 1. The implication (2) holds if and only if the set of nontangential limit points of the sequence \(\{ z_j\}\) has positive linear measure on \(| z|=1\). A point \(\zeta\) on \(| \zeta |=1\) is a nontangential limit point of \(\{z_j\}\) if for some subsequence \(z_{j_p}\) we have \[ | \zeta- z_{j_p} |= O(1- | z_{j_p} |) \qquad \text{as } p\to \infty. \]

30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
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