Extended Laguerre inequalities and a criterion for real zeros

Abstract

Let $f(z)=e^{-b{z}^2}{f}_1(z)$ where $b\ge 0$ 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.

The paper is based on a talk given at the 7th ISAAC Congress held at Imperial College in London in July 2009