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We consider a dominance order on positive vectors induced by the elementarysymmetric polynomials. Under this dominance order we provide conditions thatyield simple proofs of several monotonicity questions. Notably, our approachyields 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}). Thisinequality has been the subject of several recent articles, and only recentlyit received a full proof, albeit via a more elaborate complex-analyticapproach. We provide an elementary proof, which moreover extends to yieldsimple proofs of both old and new inequalities for R\'enyi entropy, subentropy,and quantum R\'enyi entropy.

Logarithmic inequalities under an elementary symmetric polynomial dominance order