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Theorems of Tauberian type on the distribution of the zeros of holomorphic functions. (Russian) Zbl 0545.30021
A typical theorem proved by the author: Let f(z), g(z) be holomorphic and of order \(<\alpha\) in \(\Lambda_{\alpha -\tau}=\{z: | \arg z|<\pi /2(\alpha -\tau)\} (\tau>0)\) and let all zeros of f(z)g(z) lie in \(\Lambda_{\alpha +\tau}\). Let n(r,f) denote the number of zeros of f(z) in \(\Lambda_{\alpha}\). If \(| \log f(z)-\log g(z)| =o(n(r,f))\) for \(z\in \partial \Lambda_{\alpha}\), \(| z| =r\), as \(r\to \infty\), then, under a mild growth restriction on n(r,f), \(An(r,f)<n(r,g)<Bn(r,f)\) where A,B are constants \(>0\). The paper contains many extensions and refinements of this result which cover also the cases of meromorphic functions. Some of the results were announced in Dokl. Akad. Nauk SSSR 267, 1318-1322 (1982; Zbl 0545.30018).
Reviewer: W.H.J.Fuchs

MSC:
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
40E05 Tauberian theorems, general
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