QM–AM–GM–HM inequalities


In mathematics, the QM–AM–GM–HM inequalities, also known as the mean inequality chain, state the relationship between the harmonic mean, geometric mean, arithmetic mean, and quadratic mean. Suppose that are positive real numbers. Then
In other words, QM≥AM≥GM≥HM. These inequalities often appear in mathematical competitions and have applications in many fields of science.

Proof

There are three inequalities between means to prove. There are various methods to prove the inequalities, including mathematical induction, the Cauchy–Schwarz inequality, Lagrange multipliers, and Jensen's inequality. For several proofs that GM ≤ AM, see Inequality of arithmetic and geometric means.

AM–QM inequality

From the Cauchy–Schwarz inequality on real numbers, setting one vector to :

HM–GM inequality

The reciprocal of the harmonic mean is the arithmetic mean of the reciprocals, and it exceeds by the AM-GM inequality. implies the inequality:

The ''n'' = 2 case

When n = 2, the inequalities become
which can be visualized in a semi-circle whose diameter is x1+x2.
Suppose C is a point on and let AC = x1 and BC = x2. Find the midpoint of as D and use as the center for the semi-circle from A to B. Construct perpendiculars to at D and C respectively, intersecting the circle at E and F respectively. Join and and further construct a perpendicular to 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 and Geometric mean theorem, GF to be the harmonic mean from the similarity of triangle and .