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Transactions of the American Mathematical Society

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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
DOI: https://doi.org/10.1090/S0002-9947-1981-0594422-9
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.


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

  • [1] S. Hellerstein and J. Williamson, Derivatives of entire functions 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. II, Trans. Amer. Math. Soc. 234 (1977), 497-503. MR 0481004 (58:1151)
  • [3] E. Laguerre, Sur les fonctions du genre zéro et du genre un, C. R. Acad. Sci. Paris Sér. A-B 98 (1882); Oeuvres 1 (1898), 174-177.
  • [4] B. Ja. Levin, Distribution of zeros of entire functions, Transl. Math. Monographs, vol. 5, Amer. Math. Soc., Providence, R. I., 1964. MR 0156975 (28:217)
  • [5] B. Ja. Levin and I. V. Ostrovskii, The dependence of the growth of an entire function on the distribution of the zeros of its derivatives, Amer. Math. Soc. Transl. (2) 32 (1963), 322-357.
  • [6] G. Pólya, Über Annaherung durch Polynome mit lauter reellen Wurzeln, Rend. Circ. Mat. Palermo (2) 36 (1913), 279-295.
  • [7] G. Valiron, Sur les fonctions entières d'ordre fini et d'ordre nul, et en particulier les fonctions à correspondance régulière, Ann. Fac. Sci. Univ. Toulsouse (3) 5 (1913), 117-257. MR 1508338

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DOI: https://doi.org/10.1090/S0002-9947-1981-0594422-9
Article copyright: © Copyright 1981 American Mathematical Society

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