Complex zero decreasing sequences

Abstract. The purpose of this paper is to investigate the real sequences γ0; γ1; γ2; : : : with the property that if p(x) = Pn k=0 akx k is any real polynomial, then Pn k=0 γkakx k has no more nonreal zeros than p(x). In particular, the authors establish a converse to a classical theorem of Laguerre. 1. Introduction an