On the number of real zeros of real entire functions with a non-decreasing sequence of the second quotients of Taylor coefficients

TL;DR

This paper investigates the relationship between the sequence of second quotients of Taylor coefficients of entire functions and their zeros, providing new conditions for membership in the Laguerre--Pólya class and bounds on nonreal zeros.

Contribution

It introduces new necessary conditions for functions with non-decreasing second quotient sequences to be in the Laguerre--Pólya class of type I and estimates the number of nonreal zeros.

Findings

Derived new necessary conditions for Laguerre--Pólya class membership.

Established bounds on the number of nonreal zeros.

Analyzed the impact of coefficient sequences on zero distribution.

Abstract

For an entire function $f(z) = \sum_{k=0}^\infty a_k z^k,$ $a_k >0,$ we define the sequence of the second quotients of Taylor coefficients $Q := \left( \frac{a_k^2}{a_{k-1}a_{k+1}} \right)_{k=1}^\infty$. We find new necessary conditions for a function with a non-decreasing sequence $Q$ to belong to the Laguerre--P\'olya class of type I. We also estimate the possible number of nonreal zeros for a function with a non-decreasing sequence $Q.$