When the Cauchy inequality becomes a formula

TL;DR

This paper derives a new formula for the difference between the arithmetic and geometric means of nonnegative numbers, leading to stronger inequalities and broader equality conditions, enhancing understanding of classical mean inequalities.

Contribution

It introduces a novel formula for the mean difference, resulting in stronger inequalities and broader equality cases, advancing the theory of mean inequalities.

Findings

New formula for mean difference derived

Stronger geometric-arithmetic mean inequalities established

Equality conditions extended beyond equal numbers

Abstract

In this note we revisit the classical geometric-arithmetic mean inequality and find a formula for the difference of the arithmetic and the geometric means of given $n\in\mathbb N$ nonnegative numbers $x_1,x_2,\dots,x_n$. The formula yields new stronger versions of the geometric-arithmetic mean inequality. We also find a second version of a strong geometric-arithmetic mean inequality and show that all inequalities are optimal in some sense. Anther striking novelty is, that the equality in all new inequalities holds not only in the case when all $n$ numbers are equal, but also in other cases.