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Cyclic Nevanlinna class functions in Bergman spaces


Author: Paul Bourdon
Journal: Proc. Amer. Math. Soc. 93 (1985), 503-506
MSC: Primary 30D35; Secondary 47B38
DOI: https://doi.org/10.1090/S0002-9939-1985-0774012-4
MathSciNet review: 774012
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Abstract: Let $ f$ be a function which is in both the Bergman space $ {A^p}$ $ (p \geq 1)$ and the Nevanlinna class $ N$. We show that if $ f$ is expressed as the quotient of $ {H^\infty }$ functions, then the inner part of its denominator is cyclic. As a corollary, we obtain that $ f$ is cyclic if and only if the inner part of its numerator is cyclic. These results extend those of Berman, Brown, and Cohn [2]. Using more difficult methods, they have obtained them for the case $ f \in {A^2} \cap N$. Finally, we show that the condition $ \vert f(z)\vert \geq \delta {(1 - \vert z\vert)^c}$ ($ z \in D$; $ \delta ,c$ positive constants) is sufficient for cyclicity for $ f \in {A^p} \cap N$, which answers a question of Aharonov, Shapiro, and Shields [1].


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DOI: https://doi.org/10.1090/S0002-9939-1985-0774012-4
Article copyright: © Copyright 1985 American Mathematical Society

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