New concavity and convexity results for symmetric polynomials and their ratios

Abstract

© 2018, © 2018 Informa UK Limited, trading as Taylor & Francis Group. We prove ‘power’ generalizations of Marcus–Lopes style (including McLeod and Bullen) concavity inequalities for elementary symmetric polynomials, and similar generalizations to convexity inequalities of McLeod and Baston for complete homogeneous symmetric polynomials. We also present additional concavity results for elementary symmetric polynomials, of which the main result is a concavity theorem that yields a well-known log-convexity 1972 result of Muir for positive definite matrices as a corollary.