Extended Laguerre inequalities and a criterion for real zeros

TL;DR

This paper extends classical Laguerre inequalities to provide a necessary and sufficient condition for all zeros of a certain class of entire functions to be real, enhancing understanding of zero distributions.

Contribution

It introduces extended Laguerre inequalities that characterize when all zeros of functions of the form e^{-bz^2}f_1(z) are real, generalizing classical criteria.

Findings

Provides a new set of inequalities for zero characterization

Establishes a necessary and sufficient condition for real zeros

Extends classical Laguerre inequalities to broader function classes

Abstract

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.