Inequalities via symmetric polynomial majorization
mag(2015)
摘要
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).
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