On the zeros of the derivatives of real entire functions and Wiman's conjecture (Q1119772)

The following long-standing conjecture of Wiman (1915) is proved in a quite simple and elegent way in this paper. Let f be an entire function of finite order, real on the real axis, and possessing only real zeros. Suppose that \(f''\) has no more than 2p non-real zeros for some \(p\in {\mathbb{Z}}_+\); then \[ f(z)=g(z)\exp (-az^{2p+2}), \] where \(a\geq 0\), g has genus not exceeding \(2p+1.\) If \(f(z)=\exp (\int P(z)dz)\) then the previous result yields an answer on the well-known question. Let P be a real polynomial of degree n; prove that \(P^ 2+P'\) has at most \(n+1\) real zeros. So one can add one more result to the collection of beautiful results of Pólya, Levin and Ostrovskij, Hellerstein and Williamson and others concerning this subject.