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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The zeros of the second derivative of the reciprocal of an entire function


Authors: Simon Hellerstein and Jack Williamson
Journal: Trans. Amer. Math. Soc. 263 (1981), 501-513
MSC: Primary 30D30
MathSciNet review: 594422
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Abstract: Let $ f$ be a real entire function of finite order with only real zeros. Assuming that $ f'$ has only real zeros, we show that the number of nonreal zeros of $ f''$ equals the number of real zeros of $ F''$, where $ F = 1/f$. From this, we show that $ F''$ has only real zeros if and only if $ f(z) = \exp(a{z^2} + bz + c)$, $ a \geqslant 0$, or $ f(z) = {(Az + B)^n}$, $ A \ne 0$, $ n$ a positive integer.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1981-0594422-9
PII: S 0002-9947(1981)0594422-9
Article copyright: © Copyright 1981 American Mathematical Society