Some inequalities for elementary symmetric polynomials in the complex domain

Abstract

We are investigating the problem of finding the upper bound of the modulus of elementary symmetric polynomials ek (z1, …, zn), where variables z1, …, zn are subject to conditions z1 + · · · + zn = 0 and |zj | ≤ R for all j = 1, …, n. We give a sharp upper bound in the case k = n−1. We also give an estimate in the case k = n − 2, which is sharp for even n. These estimates are then applied to give results on location of zeros of polynomials.