Some inequalities concerning symmetric forms

Abstract

Let a 1 , …, a m and b 1 , … b m be non-negative real numbers. The well-known inequality of Minkowski states that if n ≥ 1. If n is a positive integer, this inequality asserts a property of a particular symmetric form ( i.e . homogeneous polynomial) in m variables, namely the sum of the n -th powers of the variables. Some time ago, Prof. A. C. Aitken conjectured that similar properties are possessed by certain other symmetric forms. In particular, let E ( n ) ( a ) denote the n -th elementary symmetric function of a 1 , …, a m and let C ( n ) ( a ) denote the n -th complete symmetric function of a 1 , …, a m , the formal definitions being Then Prof. Aitken conjectured that