A general form of Newton-Maclaurin type inequalities

TL;DR

This work addresses extending classical Newton-Maclaurin inequalities to linear combinations of elementary symmetric means under a real-root condition on the coefficient polynomial, specifically for $S_{k;s}(x)=E_k(x)+\sum_{i=1}^s \alpha_i E_{k-i}(x)$ with $t^s+\alpha_1 t^{s-1}+\cdots+\alpha_s$ having all real roots. It proves $S_{k;s}^2(x)\ge S_{k-1;s}(x)S_{k+1;s}(x)$ for $k=s+1,\dots,n-1$ and provides equality characterizations, along with Maclaurin-type corollaries; a parallel inequality with a positive constant $\theta$ is established for $Q_{k;s}(x)=\sigma_k(x)+\sum_{i=1}^s \alpha_i \sigma_{k-i}(x)$ under Condition C. The proofs combine a real-root distribution framework for auxiliary polynomials with induction on $s$, starting from the known case $s=2$, and extend to corollaries under nonnegativity assumptions. The results have implications for fully nonlinear PDEs and geometric analysis, including operators arising in special Lagrangian-type equations.

Abstract

In this paper, we extend the classical Newton-Maclaurin inequalities to functions $S_{k;s}(x)=E_k(x)+\dsum_{i=1}^s \al_i E_{k-i}(x)$, which are formed by linear combinations of multiple basic symmetric mean. We proved that when the coefficients $\al_1,\al_2,\cdots,\al_s$ satisfy the condition that the polynomial $$t^s+\al_1 t^{s-1}+\al_2 t^{s-2}+\cdots+\al_s $$ has only real roots, the Newton-Maclaurin type inequalities hold for $S_{k;s}(x)$.

Theorems & Definitions (17)

• Theorem 1
• Remark 2
• Remark 3
• Corollary 4
• Corollary 5
• Theorem 6
• Theorem 7
• Remark 8
• Corollary 9
• Lemma 10