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

2020/01/16 by Nguyen, Thu Hien, Vishnyakova, Anna
#26C10 #30C15 #30D15 #30D35 #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2001.06302

Abstract

For an entire function f(z) = ∑_k=0^∞ a_k z^k, a_k>0, we show that if f belongs to the Laguerre-Pólya class, and the quotients q_k := \fraca_k-1²a_k-2a_k, k=2, 3, … satisfy the condition q₂ ≤ q₃, then f has at least one zero in the segment [-(a₁)/(a₂),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.

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