On the entire functions from the Laguerre-P\'olya I class having the increasing second quotients of Taylor coefficients

TL;DR

This paper establishes conditions on entire functions with positive Taylor coefficients, ensuring that most zeros are real and simple, and provides criteria for such functions to belong to the Laguerre-Pólya class of type I.

Contribution

It proves that non-decreasing second quotients of Taylor coefficients imply most zeros are real and simple, and offers a new criterion for membership in the Laguerre-Pólya class of type I.

Findings

Most zeros are real and simple under given conditions.

A criterion based on roots close to zero for belonging to the Laguerre-Pólya class.

Conditions on the sequence of Taylor coefficients influence zero distribution.

Abstract

We prove that if $f(x) = \sum_{k=0}^\infty a_k x^k,$ $a_k >0, $ is an entire function such that the sequence $Q := \left( \frac{a_k^2}{a_{k-1}a_{k+1}} \right)_{k=1}^\infty$ is non-decreasing and $\frac{a_1^2}{a_{0}a_{2}} \geq 2\sqrt[3]{2},$ then all but a finite number of zeros of $f$ are real and simple. We also present a criterion in terms of the closest to zero roots for such a function to have only real zeros (in other words, for belonging to the Laguerre--P\'olya class of type I) under additional assumption on the sequence $Q.$