On the closest to zero roots and the second quotients of Taylor coefficients of entire functions from the Laguerre-Pólya I class

Abstract

For an entire function $f(z)={\sum }_{k=0}^{\infty }{a}_k{z}^k,a_k>0,$ we show that if $f$ belongs to the Laguerre-Pólya class, and the quotients $q_k:=\frac{a_{k-1}^2}{a_{k-2}{a}_k},k=2,3,\dots$ satisfy the condition $q_2\le q_3,$ then $f$ has at least one zero in the segment $[-\frac{a_1}{a_2},0].$ We also give necessary conditions and sufficient conditions of the existence of such a zero in terms of the quotients $q_k$ for $k=2,3,4.$

18 pages. Submitted to Results in Mathematics. Partially supported by the Akhiezer Foundation. arXiv admin note: substantial text overlap with arXiv:1903.09070, arXiv:1912.10035