QM–AM–GM–HM inequalities

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In mathematics, the QM–AM–GM–HM 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 ${\displaystyle x_{1},x_{2},\ldots ,x_{n}}$ are positive real numbers. Then

${\displaystyle 0<{\frac {n}{{\frac {1}{x_{1}}}+{\frac {1}{x_{2}}}+\cdots +{\frac {1}{x_{n}}}}}\leq {\sqrt[{n}]{x_{1}x_{2}\cdots x_{n}}}\leq {\frac {x_{1}+x_{2}+\cdots +x_{n}}{n}}\leq {\sqrt {\frac {x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}}{n}}}.}$^[1]

In other words, QM≥AM≥GM≥HM. These inequalities often appear in mathematical competitions and have applications in many fields of science.^{[citation needed]}

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.

From the Cauchy–Schwarz inequality on real numbers, setting one vector to (1, 1, ...):

${\displaystyle \left(\sum _{i=1}^{n}x_{i}\cdot 1\right)^{\!2}\leq \left(\sum _{i=1}^{n}x_{i}^{2}\right)\left(\sum _{i=1}^{n}1^{2}\right)=n\,\sum _{i=1}^{n}x_{i}^{2},}$ hence ${\displaystyle \left({\frac {\sum _{i=1}^{n}x_{i}}{n}}\right)^{\!2}\leq {\frac {\sum _{i=1}^{n}x_{i}^{2}}{n}}}$. For positive ${\displaystyle x_{i}}$ the square root of this gives the inequality.

The reciprocal of the harmonic mean is the arithmetic mean of the reciprocals ${\displaystyle 1/x_{1},\dots ,1/x_{n}}$, and it exceeds ${\displaystyle 1/{\sqrt[{n}]{x_{1}\dots x_{n}}}}$ by the AM-GM inequality. ${\displaystyle x_{i}>0}$ implies the inequality:

${\displaystyle {\frac {n}{{\frac {1}{x_{1}}}+\dots +{\frac {1}{x_{n}}}}}\leq {\sqrt[{n}]{x_{1}\dots x_{n}}}.}$^[2]

When n = 2, the inequalities become

${\displaystyle {\frac {2x_{1}x_{2}}{x_{1}+x_{2}}}\leq {\sqrt {x_{1}x_{2}}}\leq {\frac {x_{1}+x_{2}}{2}}\leq {\sqrt {\frac {x_{1}^{2}+x_{2}^{2}}{2}}}}$ for all ${\displaystyle x_{1},x_{2}>0,}$ ^[3]

which can be visualized in a semi-circle whose diameter is x₁+x₂.

Suppose C is a point on [AB] and let AC = x₁ and BC = x₂. 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 △ABF is a right triangle) and Geometric mean theorem, GF to be the harmonic mean from the similarity of triangle △CGF and △DCF (whose edge [DF]'s length can be calculated using the Pythagorean theorem and the two other known edges).

• Inequalities among pythagorean means
• Generalized mean inequality

• The HM-GM-AM-QM Inequalities Archived 2022-08-14 at the Wayback Machine
• Useful inequalities cheat sheet entry "means" on the right of page 1