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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

The minimum of small entire functions


Author: P. C. Fenton
Journal: Proc. Amer. Math. Soc. 81 (1981), 557-561
MSC: Primary 30D15
DOI: https://doi.org/10.1090/S0002-9939-1981-0601729-0
MathSciNet review: 601729
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Abstract: It is shown that if $ f(z)$ is entire and satisfies $ \overline{\lim} \log M(r,f)/(\log r)^2 = \sigma < \infty$ then for a sequence of $ r \to \infty$

$\displaystyle m(r,f)/M(r,f) > \prod\limits_1^\infty \left( \frac{1 - \exp(-(2k - 1)/4\sigma )} {1 + \exp(-(2k - 1)/4\sigma)} \right)^2 + o(1).$

This proves a long-standing conjecture of P. D. Barry.

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

  • [1] J. M. Anderson, K. F. Barth and D. A. Brannan, Research problems in function theory, Bull. London Math. Soc. 9 (1977), 129-162. MR 0440018 (55:12899)
  • [2] P. D. Barry, The minimum modulus of small integral and subharmonic functions, Proc. London Math. Soc.(3) 12 (1962), 445-495. MR 0139741 (25:3172)
  • [3] E. C. Titchmarsh, Theory of functions, Oxford Univ. Press., New York, 1939.

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DOI: https://doi.org/10.1090/S0002-9939-1981-0601729-0
Article copyright: © Copyright 1981 American Mathematical Society

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