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Newton-Maclaurin type inequalities for linear combinations of elementary symmetric functions

Shuqi Hu, Changyu Ren, Ziyi Wang

Abstract

In this paper, we establish Newton-Maclaurin type inequalities for functions arising from linear combinations of primitively symmetric polynomials. This generalization extends the classical Newton-Maclaurin inequality to a broader class of functions.

Newton-Maclaurin type inequalities for linear combinations of elementary symmetric functions

Abstract

In this paper, we establish Newton-Maclaurin type inequalities for functions arising from linear combinations of primitively symmetric polynomials. This generalization extends the classical Newton-Maclaurin inequality to a broader class of functions.

Paper Structure

This paper contains 3 sections, 7 theorems, 112 equations.

Table of Contents

  1. introduction
  2. The proof of Theorem \ref{['th1']}
  3. The proof of Theorem \ref{['th5']}

Key Result

Theorem 1

For any real $\alpha, \beta\in\mathbb{R}$ and any $\boldsymbol x\in\mathbb{R}^{n}$, we have The inequality is strict unless $x_{1}=x_{2}=\cdots=x_{n},$ or unless both sides of the inequality are $0$, or unless Further, if $\alpha,\beta \geqslant 0, E_1(\boldsymbol x)\geqslant 0, E_2(\boldsymbol x)\geqslant 0$, and Then

Theorems & Definitions (11)

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