Polynomial perturbations of transcendental entire functions in the Laguerre–Pólya class

Abstract

The principal goal of this paper is to investigate and report results concerning the following problem. Determine the family of all real entire functions of positive order, φ, in the Laguerre–Polya class, such that if p is an arbitrary, non-constant real polynomial which has no zeros in common with φ, then the entire function f=φ+p possesses some non-real zeros. Ramifications of the results obtained are also considered in relation to the Hermite–Poulain Theorem and the theory of multiplier sequences.