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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 $| \zeta- z_{j_p} |= O(1- | z_{j_p} |) \qquad \text{as } p\to \infty.$

##### MSC:
 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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