On the Pólya-Wiman properties of differential operators

Let ϕ(x)=∑αnxn be a formal power series with real coefficients, and let D denote differentiation. It is shown that “for every real polynomial f there is a positive integer m0 such that ϕ(D)mf has only real zeros whenever m≥m0” if and only if “α0=0 or 2α0α2−α12<0”, and that if ϕ does not represent a Laguerre-Pólya function, then there is a Laguerre-Pólya function f of genus 0 such that for every positive integer m, ϕ(D)mf represents a real entire function having infinitely many nonreal zeros.