Hutchinson's Intervals and Entire Functions from the Laguerre-Pólya Class

Abstract

We find the intervals [α,β(α)] such that if a univariate real polynomial or entire function f(z)=a₀+a₁z+a₂z²+⋯ with positive coefficients satisfy the conditions a2_k−1ak−2ak∈[α,β(α)] for all k≥2, then f belongs to the Laguerre-Pólya class. For instance, from J.I. Hutchinson's theorem, one can observe that f belongs to the Laguerre-Pólya class (has only real zeros) when qk(f)∈[4,+∞). We are interested in finding those intervals which are not subsets of [4,+∞).