Extended Laguerre inequalities and a criterion for real zeros

摘要

Let f (z) = e~(-bz)~2 f_1 (z) where h ≥ 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.