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Transactions of the American Mathematical Society

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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.

References [Enhancements On Off] (What's this?)

  • [1] S. Hellerstein and J. Williamson, Derivatives of entire Junctions and a question of Pólya, Trans. Amer. Math. Soc. 227 (1977), 227-249. MR 0435393 (55:8353)
  • [2] -, Derivatives of entire functions and a question of Pólya, Bull. Amer. Math. Soc. 81 (1975), 453-455. MR 0361072 (50:13518)
  • [3] B. Ja. Levin, Distribution of zeros of entire functions, English transl., Transl. Math. Monographs, vol. 5, Amer. Math. Soc., Providence, R. I., 1964. MR 28 #217. MR 0156975 (28:217)
  • [4] B. Ja. Levin and I. V. Ostrovskiĭ, The dependence of the growth of an entire function on the distribution of the zeros of its derivatives, English transl., Amer. Math. Soc. Transl. (2) 32 (1963), 323-357.
  • [5] R. Nevanlinna, Le théorème de Picard-Borel et la théorie des fonctions méromorphes, Gauthier-Villars, Paris, 1929.

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