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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Bergweiler, W., Eremenko, A. & Langley, J. Real entire functions of infinite order and a conjecture of Wiman. Geom. Funct. Anal. 13, 975–991 (2003). https://doi.org/10.1007/s00039-003-0437-4
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DOI: https://doi.org/10.1007/s00039-003-0437-4