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We prove that if $f(x) = \sum_{k=0}^\infty a_k x^k,$ $a_k >0, $ is an entirefunction such that the sequence $Q := \left( \frac{a_k^2}{a_{k-1}a_{k+1}}\right)_{k=1}^\infty$ is non-decreasing and $\frac{a_1^2}{a_{0}a_{2}} \geq2\sqrt[3]{2},$ then all but a finite number of zeros of $f$ are real andsimple. We also present a criterion in terms of the closest to zero roots forsuch a function to have only real zeros (in other words, for belonging to theLaguerre--P\'olya class of type I) under additional assumption on the sequence$Q.$

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