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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Regular functions of restricted growth and their zeros in tangential regions


Author: C. N. Linden
Journal: Trans. Amer. Math. Soc. 275 (1983), 679-686
MSC: Primary 30C15; Secondary 30D15, 30D50
MathSciNet review: 682724
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Abstract: For a given function $ k$, positive, continuous, nondecreasing and unbounded on $ [0,1)$, let $ {A^{(k)}}$ denote the class of functions regular in the unit disc for which log $ \vert f(z)\vert < k(\vert z\vert)$ when $ \vert z\vert < 1$. Hayman and Korenblum have shown that a necessary and sufficient condition for the sets of positive zeros of all functions in $ {A^{(k)}}$ to be Blaschke is that

$\displaystyle \int_0^1 {\sqrt {(k(t)/(1 - t))\,dt} } $

is finite. It is shown that the imposition of a further regularity condition on the growth of $ k$ ensures that in some tangential region the zero set of each function in $ {A^{(k)}}$ is also Blaschke.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1983-0682724-9
PII: S 0002-9947(1983)0682724-9
Article copyright: © Copyright 1983 American Mathematical Society