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The majorisation principle for convex functions
Published online by Cambridge University Press: 24 February 2022
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Given positive numbers xj, yj such that 
$\sum\limits_{i = 1}^n {{x_j}} = \sum\limits_{j = 1}^n {{y_j}} $
, it can happen that 
$\sum\limits_{j = 1}^n {x_j^2} = \sum\limits_{j = 1}^n {y_j^2} $
: for example, 
$({x_j}) = (7,\,3,\,2),\,({y_j}) = (6,\,5,\,1)$
. However, such cases are exceptional.
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References
Hardy, G. H., J. Littlewood and G. Pólya, Some simple inequalities satisfied by convex functions, Messenger Math. 58 (1929) pp. 145–152.Google Scholar
Karamata, J., Sur une inégalité relative aux fonctions convexes, Publ. Math. Univ. Belgrade 1 (1932) pp. 145–148.Google Scholar
Jameson, G. J. O., Monotonicity of the midpoint and trapezium estimates for integrals, Math. Gaz. 105 (2021) pp. 433–441.10.1017/mag.2021.110CrossRefGoogle Scholar
Kadelburg, Z., Duhis, D., Luhis, M. and Matis, I., Inequalities of Karamata, Schur and Muirhead, and some applications, Teaching of Math. 8 (2005) pp. 31–45.Google Scholar
Hardy, G. H., Littlewood, J. and Pólya, G., Inequalities (2nd edn.), Cambridge University Press (1967).Google Scholar
Marshall, A. W. and Olkin, I., Inequalities: Theory of majorization and its applications, Academic Press (1979).Google Scholar
Bennett, Grahame, Some forms of majorization, Houston J. Math. 36 (2010) pp. 1037–1066.Google Scholar