Logarithmic inequalities under a 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 “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.