A general form of Newton-Maclaurin type inequalities

TL;DR

This paper generalizes classical Newton-Maclaurin inequalities to a broader class of symmetric functions formed by linear combinations of elementary symmetric means, under specific root conditions of associated polynomials.

Contribution

It introduces a unified framework extending Newton-Maclaurin inequalities to functions $S_{k;s}(x)$ with real-rooted polynomial coefficient conditions.

Findings

Inequalities hold when the polynomial $t^s + alpha t^{s-1} + eta t^{s-2} + \

The results unify various symmetric mean inequalities under a common generalization.

The conditions on polynomial roots are crucial for the validity of the inequalities.

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 $S_{k;s}(x)$.