Logarithmic inequalities under a symmetric polynomial dominance order
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Abstract:
We consider a dominance order on positive vectors induced by the elementary symmetric polynomials. Under this dominance order we provide conditions that yield simple proofs of several monotonicity questions. Notably, our approach yields a quick (4 line) proof of the so-called “sum-of-squared-logarithms” inequality conjectured in (Bîrsan, Neff, and Lankeit, J. Inequalities and Applications (2013); P. Neff, Y. Nakatsukasa, and A. Fischle; SIMAX, 35, 2014). This inequality has been the subject of several recent articles, and only recently it received a full proof, albeit via a more elaborate complex-analytic approach. We provide an elementary proof, which, moreover, extends to yield simple proofs of both old and new inequalities for Rényi entropy, subentropy, and quantum Rényi entropy.References
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Additional Information
- Suvrit Sra
- Affiliation: Laboratory for Information and Decision Systems (LIDS), Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
- MR Author ID: 815361
- Email: suvrit@mit.edu
- Received by editor(s): November 8, 2016
- Received by editor(s) in revised form: November 7, 2017
- Published electronically: November 5, 2018
- Communicated by: Walter Van Assche
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 481-486
- MSC (2010): Primary 15A42; Secondary 65F60, 26D07, 15AF5
- DOI: https://doi.org/10.1090/proc/14023
- MathSciNet review: 3894886