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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Regular functions of restricted growth and their zeros in tangential regions
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by C. N. Linden PDF
Trans. Amer. Math. Soc. 275 (1983), 679-686 Request permission

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 $|f(z)| < k(|z|)$ when $|z| < 1$. 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 \[ \int _0^1 {\sqrt {(k(t)/(1 - t)) dt} } \] 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
  • 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
  • C. N. Linden, Functions regular in the unit circle, Proc. Cambridge Philos. Soc. 52 (1956), 49–60. MR 73695
  • E. C. Titchmarsh, The theory of functions, Oxford Univ. Press, London, 1939.
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Additional Information
  • © Copyright 1983 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 275 (1983), 679-686
  • MSC: Primary 30C15; Secondary 30D15, 30D50
  • DOI: https://doi.org/10.1090/S0002-9947-1983-0682724-9
  • MathSciNet review: 682724