  Let $f(z)=e^{-bz^2}f_1(z)$ where $b \geq 0$ and $f_1(z)$ is a real entire
function of genus 0 or 1. We give a necessary and sufficient condition in terms
of a sequence of inequalities for all of the zeros of $f(z)$ to be real. These
inequalities are an extension of the classical Laguerre inequalities.
