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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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    You may be also interested in this thread: http://mathoverflow.net/questions/218811/bounding-schur-symmetric-polynomials-on-the-unit-circle2015-10-23

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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.