Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Muirhead-Rado inequality, 2: Symmetric means and inequalities

Melvyn B. Nathanson

TL;DR

The work develops a unified framework for monomial inequalities driven by symmetry, using symmetric means $[\mathbf x^{\mathbf a}]_G$ and majorization to classify when $[\mathbf x^{\mathbf b}]_G<[\mathbf x^{\mathbf a}]_G$ holds. It provides two proofs of Muirhead's inequality (and its converse) and extends the ideas to subgroups via Rado's inequality, leveraging permutohedra $K_G(\mathbf a)$ and convex separation. The approach links discrete majorization with continuous optimization and geometric objects, yielding a comprehensive theory for comparing symmetric sums of monomials in $n$ variables. This has foundational implications for inequalities across symmetric means and their generalizations to subgroup actions.

Abstract

Preliminary results from Nathanson [5] are used to prove the Muirhead and Rado inequalities.

The Muirhead-Rado inequality, 2: Symmetric means and inequalities

TL;DR

The work develops a unified framework for monomial inequalities driven by symmetry, using symmetric means and majorization to classify when holds. It provides two proofs of Muirhead's inequality (and its converse) and extends the ideas to subgroups via Rado's inequality, leveraging permutohedra and convex separation. The approach links discrete majorization with continuous optimization and geometric objects, yielding a comprehensive theory for comparing symmetric sums of monomials in variables. This has foundational implications for inequalities across symmetric means and their generalizations to subgroup actions.

Abstract

Preliminary results from Nathanson [5] are used to prove the Muirhead and Rado inequalities.
Paper Structure (5 sections, 17 theorems, 143 equations)

This paper contains 5 sections, 17 theorems, 143 equations.

Table of Contents

  1. Symmetric means of monomial functions
  2. Simple inequalities
  3. Symmetric means of functions of $n$ variables
  4. Muirhead's inequality
  5. Rado's inequality

Key Result

Theorem 1

The product of two real numbers is positive if and only if both numbers are positive or both numbers are negative.

Theorems & Definitions (32)

  • Theorem 1
  • Corollary 1
  • proof
  • Corollary 2
  • proof
  • Theorem 2
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • ...and 22 more