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

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 $k\ge 2,$ 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 $q_k(f)\in [4,+\mathrm{\infty })\mathrm{.}$ We are interested in finding those intervals which are not subsets of $[4,+\mathrm{\infty })\mathrm{.}$