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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.

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