The zeros of the second derivative of the reciprocal of an entire function

Abstract:

Let $f$ be a real entire function of finite order with only real zeros. Assuming that $f’$ has only real zeros, we show that the number of nonreal zeros of $f''$ equals the number of real zeros of $F''$, where $F = 1/f$. From this, we show that $F''$ has only real zeros if and only if $f(z) = \exp (a{z^2} + bz + c)$, $a \geqslant 0$, or $f(z) = {(Az + B)^n}$, $A \ne 0$, $n$ a positive integer.