Inequalities of Karamata, Schur and Muirhead, and some applications

Zoran Kadelburg ,Dušan Đukić ,Milivoje Lukić, Ivan Matić

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 Petrović, Steffensen and Szegö---are treated, as well.