Complex zero strip decreasing operators

Abstract

Let ϕ(z) be a function in the Laguerre–Pólya class. Write ϕ(z)=e−αz2ϕ1(z) where α≥0 and where ϕ1(z) is a real entire function of genus 0 or 1. Let f(z) be any real entire function of the form f(z)=e−γz2f1(z) where γ≥0 and f1(z) is a real entire function of genus 0 or 1 having all of its zeros in the strip S(r)={z∈C:−r≤Imz≤r}, where r>0. If αγ<1/4, the linear differential operator ϕ(D)f(z), where D denotes differentiation, is known to converge to a real entire function whose zeros also belong the strip S(r). We describe several necessary and sufficient conditions on ϕ(z) such that all zeros of ϕ(D)f(z) belong to a smaller strip S(r1)={z∈C:−r1≤Imz≤r1} where 0≤r1<r and r1 depends on ϕ(z) but is independent of f(z). We call a linear operator having this property a complex zero strip decreasing operator or CZSDO. We examine several relevant examples, in certain cases we give explicit upper and lower bounds for r′, and we state several conjectures and open problems regarding complex zero strip decreasing operators.