On the entire functions from the Laguerre-P\'olya I class having the increasing second quotients of Taylor coefficients

We prove that if $f(x)={\sum }_{k=0}^{\mathrm{\infty }}{a}_k{x}^k,$ $a_k>0,$ is an entire function such that the sequence $Q := \left( \frac{a_k^2}{a_{k-1}a_{k+1}} \right){k=1}^\infty$isnon-decreasingand$\frac{a_1^2}{a{0}a_{2}} \geq 2\sqrt[3]{2},$thenallbutafinitenumberofzerosof$f$ are real and simple. We also present a criterion in terms of the closest to zero roots for such a function to have only real zeros (in other words, for belonging to the Laguerre–P'olya class of type I) under additional assumption on the sequence $Q\mathrm{.}$