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Some inequalities concerning symmetric forms

Published online by Cambridge University Press:  26 February 2010

J. N. Whiteley
J. N. Whiteley
Affiliation:
Department of Mathematics, University College, London.
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Let a1, …, am and b1, … bm 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 a1, …, am and let C(n)(a) denote the n-th complete symmetric function of a1, …, am, the formal definitions being

Then Prof. Aitken conjectured that

Type
Research Article
Information
Mathematika , Volume 5 , Issue 1 , June 1958 , pp. 49 - 57
Copyright
Copyright © University College London 1958

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References

page 49 note * Canadian J. of Math., 9 (1957), 305–312.

page 51 note * It will be noted that, as a consequence of these results, the form δT (n) satisfies an identity similar to (12a) but with a numerical factor independent of a 1,…, a m. This property characterizes the forms T (n)(a), considered in the present paper, among a wider class of symmetric forms.

page 56 note * Hardy, Littlewood and Póya, Inequalities (1st ed.), Theorem 52.