The following problem, suggested by Laguerre’s Theorem (1884), remains open: Characterize all real sequences \(\{\mu_k\}_{k=0}^\infty\) which have the zero-diminishing property; that is, if \(p(x)=\sum_{k=0}^n a_k x^k\) is any real polynomial, then \(\sum_{k=0}^n \mu_k a_k x^k\) has no more real zeros than \(p(x)\).
In this paper this problem is solved under the additional assumption of a weak growth condition on the sequence \(\{\mu_k\}_{k=0}^\infty\), namely \( \varliminf_{n\to\infty}|\mu_n|^{1/n}< \infty\). More precisely, it is established that the real sequence \(\{\mu_k\}_{k\geq 0}\) is a weakly increasing zero-diminishing sequence if and only if there exists \( \sigma\in\{+1,-1\}\) and an entire function \[ \Phi(z)=be^{az}\prod\limits_{n\geq 1} \left (1+{x\over{ \alpha_n}}\right), a,b\in R^1, b\neq 0, \alpha_n>0, \;\forall n\geq 1, \sum\limits_{n\geq 1} {1\over{\alpha_n}}<\infty, \] such that \(\mu_k={\sigma^k}/\Phi(k), \forall k\geq 0\).