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A diagram for inequalities of symmetric functions

Published online by Cambridge University Press:  21 October 2019

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


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.

© Mathematical Association 2019 

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The Art of Problem Solving, Mildorf Inequalities, accessed June 2019, at Google Scholar
Wikipedia, Schur’s inequality (2018), available at Google Scholar