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Many inequalities regarding symmetric polynomials such as this are posed as problems

Is there an uniform method or algorithm to prove all true ones? We can assume they are input in some basis for symmetric polynomials.

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I'm thinking along the lines of using $e^{iz}$ to prove trig identities or venn diagrams to prove set theory. –  sperners lemma Nov 18 '12 at 18:35
The theory containing these any many more is decidable (en.wikipedia.org/wiki/…) but I would like a more practical algorithm which just exploits the symmetry. –  sperners lemma Nov 18 '12 at 19:14

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up vote 1 down vote accepted

There is work done in this regard and some interesting results. I recommend the following articles:

It is a fascinating area of research in which a lot remains to be discovered.

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