I have seen Muirhead's inequality being stated and used in many, many places, but I haven't ever seen a proof. Does someone have a reference or simple proof for this result?
2026-02-22 19:52:09.1771789929
Proof of Muirhead's inequality
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For non-negative integer powers we can use an induction and we can get a sum of squares.
For non-negative real powers we can use doubly stochastic matrices. See about this:
A.W.Marshal, I.Olkin "Inequalities: Theory of Majorization and Its Applicaitions",
Mathematics in science and engineering, Volume 143, 1979.