Real entire functions of infinite order and a conjecture of Wiman

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.