# New concavity and convexity results for symmetric polynomials and their ratios

## TL;DR

This paper extends classical concavity and convexity inequalities to symmetric polynomials, providing new power generalizations and implications for matrix analysis.

## Contribution

It introduces novel power-based concavity and convexity inequalities for elementary and complete homogeneous symmetric polynomials, expanding existing mathematical frameworks.

## Key findings

- New power generalizations of concavity inequalities for elementary symmetric polynomials
- Convexity inequalities for complete homogeneous symmetric polynomials
- A concavity theorem implying a known log-convexity result for positive definite matrices

## Abstract

We prove some "power" generalizations of Marcus-Lopes-style (including McLeod and Bullen) concavity inequalities for elementary symmetric polynomials, and convexity inequalities (of McLeod and Baston) for complete homogeneous symmetric polynomials. Finally, we present sundry concavity results for elementary symmetric polynomials, of which the main result is a concavity theorem that among other implies a well-known log-convexity result of Muir (1972/74) for positive definite matrices.