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Inequalities via symmetric polynomial majorization | Suvrit Sra
; | Date: |
19 Sep 2015 | Abstract: | We consider a partial order on positive vectors induced by elementary
symmetric polynomials. As a corollary we obtain a short proof of the SSLI
inequality of Neff et al. (2012), which was first obtained via a more elaborate
approach.footnote{Added 18/9/2015: It has been brought to our
attention~citep{neff.private} that a line of approach closely related to ours
has been developed by M. v{S}ilhav’y, who considers a rich generalization
based on Pick functions. This idea is natural and elegant (This idea is natural
and elegant (and was also suggested by the author to P. Neff on June 1, 2015.
Completely independent of us, M. v{S}ilhav’y has recently developed the Pick
function approach fully~citep{neff.private}), and parts of it are even
implicit in the recent article of Jozsa and Mitchison (2015). The key
contribution of v{S}ilhav’y’s work is presentation of necessary and
sufficient conditions for the E-monotonicity studied in this paper.} Our proofs
are based on a simple observation that uses suitable integral representations
and yields a family of monotonicity inequalities under a partial order
determined by elementary symmetric polynomials, and thereby yields elementary
proofs of the inequality of Neff et al. (2012) as well as related entropy
inequalities of Jozsa and Mitchison (2015). | Source: | arXiv, 1509.5902 | Services: | Forum | Review | PDF | Favorites |
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