Geometric and Functional Analysis

, Volume 13, Issue 5, pp 975–991 | Cite as

Real entire functions of infinite order and a conjecture of Wiman

  • Walter Bergweiler
  • Alexandre Eremenko
  • James K. Langley
Original Article

Abstract

We prove that if f is a real entire function of infinite order, then ff’’ has infinitely many non-real zeros. In conjunction with the result of Sheil-Small for functions of finite order this implies that if f is a real entire function such that ff’’ has only real zeros, then f is in the Laguerre-Pólya class, the closure of the set of real polynomials with real zeros. This result completes a long line of development originating from a conjecture of Wiman of 1911.

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Mathematics Subject Classification (2000).

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Copyright information

© Birkhäuser-Verlag 2003

Authors and Affiliations

  • Walter Bergweiler
    • 1
  • Alexandre Eremenko
    • 2
  • James K. Langley
    • 3
  1. 1.Mathematisches SeminarChristian-Albrechts-UniversitätKielGermany
  2. 2.Department of MathematicsPurdue UniversityWest LafayetteUSA
  3. 3.School of Mathematical SciencesUniversity of NottinghamNottinghamUK

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