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A general approach to three-variable inequalities

Published online by Cambridge University Press:  01 August 2016

H. B. Griffiths  and
A. J. Oldknow
H. B. Griffiths
Affiliation:
Chichester Institute of Higher Education, Bognor Regis PO21 1HR
A. J. Oldknow
Affiliation:
Chichester Institute of Higher Education, Bognor Regis PO21 1HR
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Extract

In tackling a problem, we needed to show that a certain 3 × 3 determinant is non-zero. This determinant could be expressed in terms of the symmetric, homogeneous function N = Σx.Σx2 + 6xyz - 2Σx3 of three variables, where, for example Σx2 = x2 + y2 + z2. We first looked into several classical books, to see if there was a relevant inequality. There was not, nor do they contain anything other than ad hoc techniques for dealing with the inequalities they give. In this paper we give a general method for dealing with a certain class of such inequalities; it might be instructive for undergraduates, partly to exercise them in applying a moderately complicated algorithm, and partly to show an application of 2- and 3- variable Calculus.

Type
Articles
Information
The Mathematical Gazette , Volume 82 , Issue 493 , March 1998 , pp. 8 - 18
Copyright
Copyright © Mathematical Association 1998 

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References

1. Griffiths, H.B. and Oldknow, A.J., Mathematics of models: continuous and discrete dynamical systems, E. Horwood, Chichester (1993).Google Scholar
2. Todhunter, I., Algebra, Macmillan, London (1866).Google Scholar