初等对称多项式控制阶下的对数不等式
Logarithmic inequalities under an elementary symmetric polynomial dominance order
摘要：
考虑了初等对称多项式在正向量上诱导的控制序。在这种优势秩序下，我们提供的条件，产生简单的证明几个单调性问题。值得注意的是，我们的方法产生了一个快速（4行）证明所谓的\emph{"sum-of-squared-logarithms"}不等式（P.~ Neff，B.~ Eidel，F.~ Osterbrink和R.~ Martin，\emph{Applied Math. \& Mechanics., 2013}; P.~内夫，Y.~ Nakatsukasa和A.~ Fischle; \emph{SIMAX, 35, 2014}）。这个不等式是最近几篇文章的主题，直到最近才得到了完整的证明，尽管是通过更复杂的复分析方法。我们提供了一个基本的证明，而且扩展到产生简单的证明新旧不等式的雷尼熵，亚熵，量子雷尼熵。
英文摘要：
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 \emph{"sum-of-squared-logarithms"} inequality conjectured in (P.~Neff, B.~Eidel, F.~Osterbrink, and R.~Martin, \emph{Applied Math. \& Mechanics., 2013}; P.~Neff, Y.~Nakatsukasa, and A.~Fischle; \emph{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.
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