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 (HM), geometric mean (GM), arithmetic mean (AM), and quadratic mean (QM; also known as root mean square). Suppose that $x_{1},x_{2},\ldots ,x_{n}$ are positive real numbers. Then

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]}

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 $(1, 1, ...)$ :

\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

\left({\frac {\sum _{i=1}^{n}x_{i}}{n}}\right)^{\!2}\leq {\frac {\sum _{i=1}^{n}x_{i}^{2}}{n}}

. For positive

x_{i}

the square root of this gives the inequality.

AM–GM inequality

Jensen's inequality states that the value of a concave function of an arithmetic mean is greater than or equal to the arithmetic mean of the function's values. Since the logarithm function is concave, we have

\log \left({\frac {\sum x_{i}}{n}}\right)\geq {\frac {1}{n}}\sum \log x_{i}={\frac {1}{n}}\log \left(\prod x_{i}\right)=\log \left(\left(\prod x_{i}\right)^{1/n}\right).

Taking antilogs of the far left and far right sides, we have the AM–GM inequality.

HM–GM inequality

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

{\frac {n}{{\frac {1}{x_{1}}}+\dots +{\frac {1}{x_{n}}}}}\leq {\sqrt[{n}]{x_{1}\dots x_{n}}}.

^[2]

The n = 2 case

When n = 2, the inequalities become

{\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

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).

References

^ Djukić, Dušan (2011). The IMO compendium: a collection of problems suggested for the International Mathematical Olympiads, 1959-2009. Problem books in mathematics. International mathematical olympiad. New York: Springer. p. 7. ISBN 978-1-4419-9854-5.
^ Sedrakyan, Hayk; Sedrakyan, Nairi (2018), Sedrakyan, Hayk; Sedrakyan, Nairi (eds.), "The HM-GM-AM-QM Inequalities", Algebraic Inequalities, Problem Books in Mathematics, Cham: Springer International Publishing, p. 23, doi:10.1007/978-3-319-77836-5_3, ISBN 978-3-319-77836-5, retrieved 2023-11-26
^ Sedrakyan, Hayk; Sedrakyan, Nairi (2018), Sedrakyan, Hayk; Sedrakyan, Nairi (eds.), "The HM-GM-AM-QM Inequalities", Algebraic Inequalities, Problem Books in Mathematics, Cham: Springer International Publishing, p. 21, doi:10.1007/978-3-319-77836-5_3, ISBN 978-3-319-77836-5, retrieved 2023-11-26

External links

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

[1] Djukić, Dušan (2011). The IMO compendium: a collection of problems suggested for the International Mathematical Olympiads, 1959-2009. Problem books in mathematics. International mathematical olympiad. New York: Springer. p. 7. ISBN 978-1-4419-9854-5.

[2] Sedrakyan, Hayk; Sedrakyan, Nairi (2018), Sedrakyan, Hayk; Sedrakyan, Nairi (eds.), "The HM-GM-AM-QM Inequalities", Algebraic Inequalities, Problem Books in Mathematics, Cham: Springer International Publishing, p. 23, doi:10.1007/978-3-319-77836-5_3, ISBN 978-3-319-77836-5, retrieved 2023-11-26

[3] Sedrakyan, Hayk; Sedrakyan, Nairi (2018), Sedrakyan, Hayk; Sedrakyan, Nairi (eds.), "The HM-GM-AM-QM Inequalities", Algebraic Inequalities, Problem Books in Mathematics, Cham: Springer International Publishing, p. 21, doi:10.1007/978-3-319-77836-5_3, ISBN 978-3-319-77836-5, retrieved 2023-11-26

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