Each mean has the following properties:
- Value preservation:
- First order homogeneity:
- Invariance under exchange: for any and .
These means were studied with proportions by Pythagoreans and later generations of Greek mathematicians (Thomas Heath, History of Ancient Greek Mathematics) because of their importance in geometry and music. The harmonic and arithmetic means are reciprocal duals of each other for positive arguments () while the geometric mean is its own reciprocal dual.
Inequalities among means
There is an ordering to these means (if all of the are positive)
with equality holding if and only if the are all equal. This is a generalization of the inequality of arithmetic and geometric means and a special case of an inequality for generalized means. The proof follows from the artithmetic-geometric mean inequality, , and reciprocal duality ( and are also reciprocal dual to each other).
The study of the pythagorean means is closely related to the study of majorization and Schur-convex functions. The harmonic and geometric means are concave symmetric functions of their arguments, and hence Schur-concave, while the arithmetic mean is a linear function of its arguments, so both concave and convex.