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06019065 Kova\v{c}ec, Alexander Kuhlmann, Salma Riener, Cordian A note on extrema of linear combinations of elementary symmetric functions. Linear Multilinear Algebra 60, No. 2, 219-224 (2012). 2012 Taylor \& Francis, Abingdon EN extrema elementary symmetric functions hyperbolic polynomials

doi:10.1080/03081087.2011.588438

Summary: 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.