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Muirhead's Theorem

A necessary and sufficient condition that [alpha^'] should be comparable with [alpha] for all positive values of the a is that one of (alpha^') and (alpha) should be majorized by the other. If (alpha^')≺(alpha), then

[alpha^']<=[alpha],

with equality only when (alpha^') and (alpha) are identical or when all the a are equal. See Hardy et al. (1988) for a definition of notation.

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References

Hardy, G. H.; Littlewood, J. E.; and Pólya, G. "Muirhead's Theorem" and "Proof of Muirhead's Theorem." §2.18 and 2.19 in Inequalities, 2nd ed. Cambridge, England: Cambridge University Press, pp. 44-48, 1988.Muirhead, R. F. "Some Methods Applicable to Identities and Inequalities of Symmetric Algebraic Functions of n Letters." Proc. Edinburgh Math. Soc. 21, 144-157, 1903.

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Muirhead's Theorem

Cite this as:

Weisstein, Eric W. "Muirhead's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MuirheadsTheorem.html

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