# Means Moments and Newton's Inequalities

## TL;DR

This paper explores how Newton's and Maclaurin's inequalities refine classical mean inequalities, providing new bounds involving means, variance, and higher moments of positive real numbers.

## Contribution

It introduces new inequalities involving third and fourth central moments, extending classical mean inequalities with refined bounds.

## Key findings

- Newton's inequalities refine AM-GM-HM inequalities.
- New inequalities involving third and fourth moments are derived.
- Results provide tighter bounds on means and moments.

## Abstract

It is shown that Newton's inequalities and the related Maclaurin's inequalities provide several refinements of the fundamental Arithmetic mean - Geometric mean - Harmonic mean inequality in terms of the means and variance of positive real numbers. We also obtain some inequalities involving third and fourth central moments of real numbers.