On the zeros of analytic functions in the disk with a given majorant near its boundary. (English. Russian original) Zbl 1177.30039
Math. Notes 85, No. 2, 274-287 (2009); translation from Mat. Zametki 85, No. 2, 300-312 (2009).
Summary: Suppose that \(\lambda \) is an arbitrary positive function from \(C[0,1)\) such that \(\lambda (r) \rightarrow \infty \) as \(r \rightarrow 1 - 0\) and some growth regularity conditions are satisfied, \(A(\lambda )\) is the set of all holomorphic functions \(f\) in the unit disk for which \(\ln |f(z)| \leq c \cdot \lambda (|z|)\), \( |z|< 1\). In this paper, we establish that there exists a function \(f \in A(\lambda )\) with the root set \(\{z_k\}_{k=1}^{+\infty }\) such that the sequence \(\{|z_k|\}_{k=1}^{+\infty }\) is the uniqueness set for the class \(A(\lambda )\).
MSC:
| 30D35 | Value distribution of meromorphic functions of one complex variable, Nevanlinna theory |
Keywords:
entire function; meromorphic function; root set; uniqueness set; Nevanlinna characteristic; Blaschke conditionReferences:
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