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 $(\alpha')\prec(\alpha)$, then

$$[\alpha']\leq [\alpha],$$

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.

References

Hardy, G. H.; Littlewood, J. E.; and Pólya, G. Inequalities, 2nd ed. Cambridge, England: Cambridge University Press, pp. 44-48, 1988.