Hutchinson's intervals and entire functions from the Laguerre-Pólya class

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--P\'olya class. For instance, from J.I.~Hutchinson's theorem, one can observe that $f$ belongs to the Laguerre--P\'olya class (has only real zeros) when $q_k(f) \in [4, + \infty).$ We are interested in finding those intervals which are not subsets of $[4, + \infty).$