An Arithmetic and Geometric Mean Invariant

TL;DR

This paper proves that both arithmetic and geometric means maintain constant ratios across sub-intervals on a log scale, revealing an invariance property with potential implications for mathematical analysis.

Contribution

It introduces and proves an invariance property of arithmetic and geometric means on positive real intervals, including a continuous analog.

Findings

Arithmetic and geometric means produce constant ratios on a log scale.

The invariance property extends to continuous intervals.

Potential applications in mathematical analysis and related fields.

Abstract

A positive real interval, [a, b], can be partitioned into sub-intervals such that sub-interval widths divided by sub-interval "average" values remains constant. That both Arithmetic Mean and Geometric Mean "average" values produce constant ratios for the same log scale is the stated invariance proved in this short note. The continuous analog is briefly considered and shown to have similar properties.