The operators preserving real-rootedness and inequalities for symmetric function

In this paper, we will demonstrate two unified inequalities for the elementary symmetric function via the theory of the operators preserving real-rootedness. Utilizing this approach, we prove the convexity of the ratio of the elementary symmetric function and give an affirmative answer to Briggs' conjecture. And we consider the Newton sums S k (x) and show that the sequence S k (x) k ≥ 0 alternately satisfies the Laguerre inequality of any order. This result mirrors a celebrated theorem of Hermite on the relation between the Hermite matrix and the real-rootedness of polynomials. Moreover, we apply our approach to some inequalities arising from the study of the Hessian equation. We provide an alternative proof of a series of inequalities given by Ren. At last, we also give a generalization of Tao's Maclaurin type inequality. Furthermore, we partially settle Wagner's conjecture on shifted multiplier sequences and prove that all order difference of the partition function are shifted multiplier sequences.