Differential operators and entire functions with simple real zeros

Let φ and f be functions in the Laguerre–Pólya class. Write φ(z) = e −αz2 φ1(z) and f (z) = e −βz2 f1(z), where φ1 and f1 have genus 0 or 1 and α,β 0. If αβ < 1/4 and φ has infinitely many zeros, then φ(D)f (z) has only simple real zeros, where D denotes differentiation.  2004 Elsevier Inc. All rights reserved.