Derivatives of entire functions and a question of Pólya. II

Abstract:

It is shown that if f is an entire function of infinite order, which is real on the real axis and has, along with $f’$, only real zeros, then $f''$ has nonreal zeros (in fact, infinitely many). The finite order case was treated by the authors in a preceding paper. The combined results show that the only real entire functions f for which $f,f’$, and $f''$ have only real zeros are those in the Laguerre-Pólya class, i.e. \[ f(z) = {z^m}\exp \{ - a{z^2} + bz + c\} \prod \limits _n {\left ( {1 - \frac {z}{{{z_n}}}} \right )} {e^{z/{z_n}}},\] $a \geqslant 0,b,c$ and the ${z_n}$ real, and $\Sigma z_n^{ - 2} < \infty$. This gives a strong affirmative version of an old conjecture of Pólya.