A geometric construction of the Quadratic mean and the Pythagorean means (of two numbers a and b). Harmonic mean denoted by H, Geometric by G, Arithmetic by A and Quadratic mean (also known as Root mean square) denoted by Q.
Comparison of the arithmetic, geometric and harmonic means of a pair of numbers. The vertical dashed lines are asymptotes for the harmonic means.
These means were studied with proportions by Pythagoreans and later generations of Greek mathematicians 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.
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.