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

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Date
2022-12-12MFO Scientific Program
OWLF 2022Series
Oberwolfach Preprints;2022-19Author
Nguyen, Thu Hien
Vishnyakova, Anna
Metadata
Show full item recordOWP-2022-19
Abstract
We find the intervals $[\alpha, \beta (\alpha)]$ such that if a univariate real polynomial or entire function $f(z) = a_0 + a_1 z + a_2 z^2 + \cdots $ with positive coefficients satisfy the conditions $ \frac{a_{k-1}^2}{a_{k-2}a_{k}} \in [\alpha, \beta(\alpha)]$ for all $k \geq 2,$ 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 $q_k(f) \in [4, + \infty).$ We are interested in finding those intervals which are not subsets of $[4, + \infty).$