Logarithmic inequalities under an elementary symmetric polynomial dominance order

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 R\'enyi entropy, subentropy, and quantum R\'enyi entropy.