Logarithmic inequalities under a symmetric polynomial dominance order
Logarithmic inequalities under a symmetric polynomial dominance order
复制标题
对称多项式优势阶下的对数不等式
DOI：
10.1090/proc/14023
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发表时间：
2017
影响因子：
1
通讯作者：
S. Sra
中科院分区：
文献类型：
--
作者：
S. Sra
我们考虑由初等对称多项式诱导的正向量上的一个占优序。在这种占优序下，我们给出了几个单调性问题的简单证明条件。值得注意的是，我们的方法给出了一个快速(4行)的证明，证明了(布尔桑，内夫和兰凯特，J.不等式和应用(2013)；P.内夫，Y.Nakatsukasa和A.Fischle；Simax，35,2014)中猜想的所谓的“平方和-对数”不等式。这种不平等是最近几篇文章的主题，直到最近，它才得到了充分的证明，尽管是通过更复杂的分析方法。我们给出了一个初等证明，并且推广到给出了关于Rényi熵、子熵和量子Rényi熵的新旧不等式的简单证明。参考文献
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