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A diagram for inequalities of symmetric functions

Published online by Cambridge University Press:  21 October 2019

Stan Dolan*
Affiliation:
126A Harpenden Road, St Albans AL3 6BZ

Extract

Consider the problem of proving that, for any positive numbers x, y and z,

$${\rm{9 (}}{{\rm{x}}^{\rm{3}}}{\rm{ + }}{{\rm{y}}^{\rm{3}}}{\rm{ + }}{{\rm{z}}^{\rm{3}}}{\rm{) }} \ge {\rm{ (x + y + z}}{{\rm{)}}^{\rm{3}}}{\rm{ }}{\rm{.}}$$

This is an example of a type of inequality that frequently occurs in Olympiad-style problems, [1]. These problems may involve symmetric functions of more or fewer variables than the three used here. However, three variables are commonly used and appear to give appropriately difficult problems without making excessive computational demands.

Type
Articles
Copyright
© Mathematical Association 2019 

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References

The Art of Problem Solving, Mildorf Inequalities, accessed June 2019, at https://artofproblemsolving.com/articles/files/MildorfInequalities.pdf Google Scholar
Wikipedia, Schur’s inequality (2018), available at https://en.wikipedia.org/wiki/Schur%27s_inequality Google Scholar