Sharp bounds for the difference between the arithmetic and geometric means

TL;DR

This paper establishes precise bounds on the difference between the weighted arithmetic mean and the geometric mean of positive numbers, expressed through the variance of their square roots.

Contribution

It introduces sharp bounds for the difference between the weighted arithmetic and geometric means based on the variance of the square roots of the variables.

Findings

Derived exact bounds for the mean difference

Expressed bounds in terms of variance of square roots

Enhanced understanding of inequalities between means

Abstract

We present sharp bounds for $\sum_{i=1}^n \alpha_i x_i - \prod_{i=1}^n x_i^{\alpha_i}$ in terms of the variance of the vector $(x_1^{1/2},...,x_n^{1/2})$.