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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.

References [Enhancements On Off] (What's this?)

  • [1] W. K. Hayman and B. Korenblum, A critical growth rate for functions regular in a disk, Michigan Math. J. 27 (1980), no. 1, 21–30. MR 555833
  • [2] C. N. Linden, Functions regular in the unit circle, Proc. Cambridge Philos. Soc. 52 (1956), 49–60. MR 0073695
  • [3] E. C. Titchmarsh, The theory of functions, Oxford Univ. Press, London, 1939.

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DOI: https://doi.org/10.1090/S0002-9947-1983-0682724-9
Article copyright: © Copyright 1983 American Mathematical Society