We find 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 satisfies the conditions \( q_k(f) = {a_{k-1}^2}/({a_{k-2}a_{k}}) \in [\alpha , \beta (\alpha )]\) for all \(k \geqslant 2\) , then f belongs to the Laguerre–Pólya class. For instance, from 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 \([4, + \infty )\) .
