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Transactions of the American Mathematical Society

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



Derivatives of entire functions and a question of Pólya. II

Authors: Simon Hellerstein and Jack Williamson
Journal: Trans. Amer. Math. Soc. 234 (1977), 497-503
MSC: Primary 30A66
MathSciNet review: 0481004
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Abstract: It is shown that if f is an entire function of infinite order, which is real on the real axis and has, along with $ f'$, only real zeros, then $ f''$ has nonreal zeros (in fact, infinitely many). The finite order case was treated by the authors in a preceding paper. The combined results show that the only real entire functions f for which $ f,f'$, and $ f''$ have only real zeros are those in the Laguerre-Pólya class, i.e.

$\displaystyle f(z) = {z^m}\exp \{ - a{z^2} + bz + c\} \prod\limits_n {\left( {1 - \frac{z}{{{z_n}}}} \right)} {e^{z/{z_n}}},$

$ a \geqslant 0,b,c$ and the $ {z_n}$ real, and $ \Sigma z_n^{ - 2} < \infty $. This gives a strong affirmative version of an old conjecture of Pólya.

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Article copyright: © Copyright 1977 American Mathematical Society

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