On the number of real zeros of real entire functions with a non-decreasing sequence of the second quotients of Taylor coefficients

Abstract

For an entire function $f(z)={\sum }_{k=0}^{\infty }{a}_k{z}^k,$ $a_k>0,$ we define the sequence of the second quotients of Taylor coefficients $Q:={\left(\frac{a_k^2}{a_{k-1}{a}_{k+1}}\right)}_{k=1}^{\infty }$. We find new necessary conditions for a function with a non-decreasing sequence $Q$ to belong to the Laguerre--Pólya class of type I. We also estimate the possible number of nonreal zeros for a function with a non-decreasing sequence $Q.$

18 pages. arXiv admin note: text overlap with arXiv:2008.04754, arXiv:2001.06302