On the necessary condition for entire function with the increasing second quotients of Taylor coefficients to belong to the Laguerre-P\'olya 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-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 $q_\infty$ $(q_\infty\approx 3{.}2336) .$