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Transactions of the American Mathematical Society

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

 

 

The minimum modulus of certain small entire functions


Author: P. C. Fenton
Journal: Trans. Amer. Math. Soc. 271 (1982), 183-195
MSC: Primary 30D15
DOI: https://doi.org/10.1090/S0002-9947-1982-0648085-5
MathSciNet review: 648085
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Abstract: Suppose that $ f(z)$ is an entire function satisfying

$\displaystyle \mathop {\min }\limits_{\vert z\vert = r} \vert f(z)\vert \leqslant C(\sigma )\mathop {\max }\limits_{\vert z\vert = r} \vert f(z)\vert,$

for $ r \geqslant {\rho _0} > 0$, where $ \sigma > 0$ and

$\displaystyle C(\sigma ) = \prod\limits_{k = 1}^\infty {{{\left\{ {\frac{{1 - \... ...- (2k - 1) / 4\sigma )}} {{1 + \exp ( - (2k - 1) / 4\sigma )}}} \right\}}^2}.} $

It is shown that

$\displaystyle \mathop {\underline {\lim } }\limits_{r \to \infty } \frac{{{{\ma... ...)}^2}}} {{\log r}} \geqslant - 2\sigma \log (\max ({\rho _0},\vert{a_1}\vert)),$

where $ {a_1}$ is the first nonzero zero of $ f$.

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DOI: https://doi.org/10.1090/S0002-9947-1982-0648085-5
Article copyright: © Copyright 1982 American Mathematical Society