The convexity and the concavity derived from Newton's inequality

Abstract

By Newton’s inequality, a sequence {ai}i=0 of nonnegative real numbers is unimodal if its generating function ∑i=0 aix has only real zeros. This paper is devoted to show that there exist two indices s and t with s t , such that a0,a1, . . . ,as−1,as and at ,at+1, . . . ,an are convex, while as−1,as, . . . ,at ,at+1 is concave. Mathematics subject classification (2010): 05A10, 05A20.