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Muirhead's theorem (Muirhead's inequality)¤Ï²¼µ­¤ÎÄ̤ê

Let s1 &#8805; s2 &#8805; ... &#8805; sn &#8805; 0, and
t1 &#8805; t2 &#8805; ... &#8805; tn &#8805; 0, and
¦²si = ¦²ti (i=1¡Án)
and for all k < n, ¦²si &#8805; ¦²ti (i=1¡Ák)

Then for all nonnegative numbers x1, x2, ..., xn,
¦²x1^s¦Ò(1) x2^s¦Ò(2) ... xn^s¦Ò(n) &#8805; ¦²x1^t¦Ò(1) x2^t¦Ò(2) ... xn^t¦Ò(n)
where the sums run over all the permutations ¦Ò of {1, 2, 3, ..., n}¡¡¢¨
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Muirhead's theorem (Muirhead's inequality)¤Ï²¼µ­¤ÎÄ̤ê

Let s1 >= s2 >= ... >= sn >= 0, and
t1 >= t2 >= ... >= tn >= 0, and
¦²si = ¦²ti (i=1¡Án)
and for all k < n, ¦²si >= ¦²ti (i=1¡Ák)

Then for all nonnegative numbers x1, x2, ..., xn,
¦²x1^s¦Ò(1) x2^s¦Ò(2) ... xn^s¦Ò(n) >= ¦²x1^t¦Ò(1) x2^t¦Ò(2) ... xn^t¦Ò(n)
where the sums run over all the permutations ¦Ò of {1, 2, 3, ..., n}¡¡¢¨
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