Geodesic
Inequalities of Karamata, Schur and Muirhead, and some applications
The Teaching of Mathematics, VIII (2005) no. 1, p. 31 .

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Three classical general inequalities---those of Karamata, Schur and Muirhead---are proved in this article. They can be used in proving other inequalities, particularly those appearing as problems in mathematical competitions, including International Mathematical Olympiads. Some problems of this kind are given as examples. Several related inequalities---those of Petrovi�, Steffensen and Szeg�---are treated, as well.
Classification : 00A35 H34
Keywords: Relation of majorization, divided difference, Karamata's inequality, Petrović's inequality, Steffensen's inequality, Schur's inequality, Muirhead's inequality. 
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Zoran Kadelburg; Dušan Đukić; Milivoje Lukić; Ivan Matić. Inequalities of Karamata, Schur and Muirhead, and some applications. The Teaching of Mathematics, VIII (2005) no. 1, p. 31 . http://geodesic.mathdoc.fr/item/TM2_2005_VIII_1_a2/