  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 $q_\infty$ $(q_\infty\approx 3{.}2336) .$
