A Note on Extrema of Linear Combinations of Elementary Symmetric Functions

TL;DR

This paper introduces a new approach to extremal problems involving elementary symmetric functions, generalizing previous results and critiquing earlier proofs.

Contribution

It presents a novel method for deriving extremal results on elementary symmetric functions and establishes a more general theorem that encompasses prior findings.

Findings

New proof technique for elementary symmetric functions

Generalization of previous extremal results

Identification of flaws in earlier proofs

Abstract

This note provides a new approach to a result of Foregger and related earlier results by Keilson and Eberlein. Using quite different techniques, we prove a more general result from which the others follow easily. Finally, we argue that the proof given by Foregger is flawed.