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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Successive derivatives of entire functions


Authors: Simon Hellerstein and Jack Williamson
Journal: Proc. Amer. Math. Soc. 66 (1977), 105-108
MSC: Primary 30A66
MathSciNet review: 0460637
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Abstract: We show that if f is a real entire function which has, along with each of its derivatives, only real nonpositive zeros, then either $ f(z) = c{e^{\sigma z}},c$ and $ \sigma $ real constants, or

$\displaystyle f(z) = c{z^m}{e^{\sigma z}}\prod\limits_n {\left( {1 + \frac{z}{{\vert{z_n}\vert}}} \right)} $

where $ \sigma \geqslant 0$ and $ \sum\nolimits_n {\vert{z_n}{\vert^{ - 1}} < \infty } $.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1977-0460637-8
PII: S 0002-9939(1977)0460637-8
Keywords: Real entire function, derivative, genus of a canonical product
Article copyright: © Copyright 1977 American Mathematical Society