# On the necessary condition for entire function with the increasing second quotients of Taylor coefficients to belong to the Laguerre-P\'olya class

## TL;DR

This paper establishes a necessary condition involving the increasing second quotients of Taylor coefficients for an entire function with positive coefficients to belong to the Laguerre-Pólya class, identifying a critical limit constant.

## Contribution

It introduces a new necessary condition based on the behavior of second quotients of Taylor coefficients for classifying entire functions within the Laguerre-Pólya class.

## Key findings

- Functions with increasing second quotients do not belong to the Laguerre-Pólya class if the limit is below a specific constant.
- Identifies a critical limit constant approximately equal to 3.2336.
- Provides a criterion involving the asymptotic behavior of Taylor coefficient quotients.

## Abstract

For an entire function $f(z) = \sum_{k=0}^\infty a_k z^k, a_k>0,$ we show that $f$ does not belong to the Laguerre-P\'olya class if the quotients $\frac{a_{n-1}^2}{a_{n-2}a_n}$ are increasing in $n$, and $c:= \lim\limits_{n\to \infty} \frac{a_{n-1}^2}{a_{n-2}a_n}$ is smaller than an absolute constant $q_\infty$ $(q_\infty\approx 3{.}2336) .$