On the necessary condition for entire function with the increasing second quotients of Taylor coefficients to belong to the Laguerre-Pólya class

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-Pólya class if the quotients $\frac{a_{n-1}^2}{a_{n-2}{a}_n}$ are increasing in $n$, and $c:=\underset{n\to \infty }{\lim }\frac{a_{n-1}^2}{a_{n-2}{a}_n}$ is smaller than an absolute constant $q_{\infty }$ $(q_{\infty }\approx 3.2336).$