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

Abstract

We prove that if $f(x)={\sum }_{k=0}^{\infty }{a}_k{x}^k,$ $a_k>0,$ is an entire function such that the sequence $Q:={\left(\frac{a_k^2}{a_{k-1}{a}_{k+1}}\right)}_{k=1}^{\infty }$ is non-decreasing and $\frac{a_1^2}{a_0{a}_2}\ge 2\sqrt[3]{2},$ then all but a finite number of zeros of $f$ 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--Pólya class of type I) under additional assumption on the sequence $Q.$

16 pages. arXiv admin note: text overlap with arXiv:2001.06302, arXiv:1912.10035, arXiv:1903.09070