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A uniqueness theorem for bounded analytic functions. (English) Zbl 0889.30005
A nonincreasing function \(g:[0,1)\to(0,1/e]\) is said to be an essential minorant for \(H^\infty\) if for every separated non-Blaschke sequence \(\lambda_k\), \(f\in H^\infty\) and \[ |f(\lambda_k)|\leq g(|\lambda_k|) \] for all \(k\) imply \(f(z)\equiv 0\). The authors prove the following theorem.
Theorem: A nonincreasing function \(g:[0,1)\to(0,1/e]\) is an essential minorant for \(H^\infty\) if and only if \[ \int^1_0\frac{dr}{(1-r)\log(1/g(r))}<+\infty. \]

MSC:
30A99 General properties of functions of one complex variable
31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
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