A note on extrema of linear combinations of elementary symmetric functions

Abstract

This note provides a new approach to a result of Foregger [T.H. Foregger, On the relative extrema of a linear combination of elementary symmetric functions, Linear Multilinear Algebra 20 (1987) pp. 377–385] and related earlier results by Keilson [J. Keilson, A theorem on optimum allocation for a class of symmetric multilinear return functions, J. Math. Anal. Appl. 15 (1966), pp. 269–272] and Eberlein [P.J. Eberlein, Remarks on the van der Waerden conjecture, II, Linear Algebra Appl. 2 (1969), pp. 311–320]. Using quite different techniques, we prove a more general result from which the others follow easily. Finally, we argue that the proof in [Foregger, 1987] is flawed.

Keywords:

• extrema
• elementary symmetric functions
• hyperbolic polynomials

AMS Subject Classifications::

• Primary: 14P99, 14Q99
• Secondary: 26C05, 26C10

Acknowledgement

We thank António Leal Duarte for having drawn our attention to Kato's book.