Successive derivatives of entire functions

Abstract:

We show that if f is a real entire function which has, along with each of its derivatives, only real nonpositive zeros, then either $f(z) = c{e^{\sigma z}},c$ and $\sigma$ real constants, or \[ f(z) = c{z^m}{e^{\sigma z}}\prod \limits _n {\left ( {1 + \frac {z}{{|{z_n}|}}} \right )} \] where $\sigma \geqslant 0$ and $\sum \nolimits _n {|{z_n}{|^{ - 1}} < \infty }$.