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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
##### Keywords:
uniqueness theorem; essential minorant
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