In mathematics, the QMâÂÂAMâÂÂGMâÂÂHM inequalities, also known as the mean inequality chain, state the relationship between the harmonic mean (HM), geometric mean (GM), arithmetic mean (AM), and quadratic mean (QM; also known as root mean square). Suppose that are positive real numbers. Then
In other words, QMâÂÂ¥AMâÂÂ¥GMâÂÂ¥HM. These inequalities often appear in mathematical competitions and have applications in many fields of science.
There are three inequalities between means to prove. There are various methods to prove the inequalities, including mathematical induction, the CauchyâÂÂSchwarz inequality, Lagrange multipliers, and Jensen's inequality. For several proofs that GM ⤠AM, see Inequality of arithmetic and geometric means.
From the CauchyâÂÂSchwarz inequality on real numbers, setting one vector to :
The reciprocal of the harmonic mean is the arithmetic mean of the reciprocals , and it exceeds by the AM-GM inequality. implies the inequality:
When n = 2, the inequalities become
which can be visualized in a semi-circle whose diameter is x<sub>1</sub>+x<sub>2</sub>.
Suppose C is a point on [AB] and let AC = x<sub>1</sub> and BC = x<sub>2</sub>. Find the midpoint of [AB] as D and use as the center for the semi-circle from A to B. Construct perpendiculars to [AB] at D and C respectively, intersecting the circle at E and F respectively. Join [CE] and [DF] and further construct a perpendicular [CG] to [DF] at G. The length of DE is the arithmetic mean by the virtue of being the ray of the circle. CE can be calculated to be the quadratic mean from the Pythagorean theorem, CF to be the geometric mean from a combination of Thales's theorem (establishing that is a right triangle) and Geometric mean theorem, GF to be the harmonic mean from the similarity of triangle and (whose edge [DF]'s length can be calculated using the Pythagorean theorem and the two other known edges).