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The reciprocal of an entire function of infinite order and the distribution of the zeros of its second derivative

Author: John Rossi
Journal: Trans. Amer. Math. Soc. 270 (1982), 667-683
MSC: Primary 30D35
MathSciNet review: 645337
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Abstract: Let $ f$ be a real entire function of infinite order whose zeros together with those of $ f'$ are all real. It is proved that $ (1/f)''$ has an infinity of nonreal zeros. The location of the zeros of $ f''$ and $ (1/f)''$ is also investigated. The result complements a finite order result of Hellerstein and Williamson.

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

  • [1] A. Edrei, Meromorphic functions with three radially distributed values, Trans. Amer. Math. Soc. 78 (1955), 276-293. MR 0067982 (16:808d)
  • [2] W. K. Hayman, Meromorphic functions, Oxford Univ. Press, Oxford, 1964. MR 0164038 (29:1337)
  • [3] 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)
  • [4] -, Derivatives of entire functions and a question of Pólya. II, Trans. Amer. Math. Soc. 234 (2) (1977), 497-503. MR 0481004 (58:1151)
  • [5] -, The zeros of the second derivative of the reciprocal of an entire function, Trans. Amer. Math. Soc. 263 (1981), 501-513. MR 594422 (81k:30036)
  • [6] B. Ja. Levin, Distribution of zeros of entire functions, GITTL, Moscow, 1956; English transl., Transl. Math. Monographs, vol. 5, Amer. Math. Soc., Providence, R. I., 1964. MR 0156975 (28:217)
  • [7] B. Ja. Levin and I. V. Ostrovskiĭ, On the dependence of the growth of an entire function on the distribution of the zeros of its derivatives (on a question of G. Pólya and A. Wiman), Sibirsk. Mat. Ž. 1 (1960), 427-455; English transl., Amer. Math. Soc. Transl. (2) 32 (1963), 323-357. MR 0130979 (24:A833)
  • [8] G. Pólya, Sur une question concernant les fonctions entières, C. R. Acad. Sci. Paris 158 (1914), 320-333.
  • [9] -, Bemerkung zur Theorie der ganzen Funktionen, Jber. Deutsch. Math.-Verein 24 (1915), 392-400.
  • [10] -, On the zeros of the derivatives of a function and its analytic character, Bull. Amer. Math. Soc. 49 (1943), 178-191. MR 0007781 (4:192d)
  • [11] J. Rossi, Two problems in entire function theory: Reciprocals of entire functions of infinite order and functions extremal for the $ \cos \beta \lambda $ theorem, Doctoral dissertation, University of Hawaii, 1980.
  • [12] W. Rudin, Real and complex analysis, 2nd ed., McGraw-Hill, New York, 1974. MR 0344043 (49:8783)

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Keywords: Real entire function, real meromorphic function, infinite order
Article copyright: © Copyright 1982 American Mathematical Society

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