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.
Fan et al. (Tue,) studied this question.