On the entire functions from the Laguerre–Pólya I class having the increasing second quotients of Taylor coefficients

We prove that if f ( x ) = ∑ k 0 ∞ a , > is an entire function such the sequence Q : 2 − 1 + non-decreasing and ≥ 3 then all but finite number of zeros are real simple. also present criterion in terms closest to zero roots for have only (in other words, belonging Laguerre–Pólya class type I) under additional assumption on .