Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 30D30

Retrieve articles in all journals with MSC: 30D30

Additional Information

Article copyright: © Copyright 1981 American Mathematical Society

American Mathematical Society