View source for QM-AM-GM-HM inequalities

{{refimprove|date=November 2023}}
{{Short description|Mathematical relationships}}
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 [[Root mean square|quadratic mean]] (also known as root mean square). Suppose that <math>x_1, x_2, \ldots, x_n</math> are positive [[real number]]s. Then

: <math>0<\frac{n}{\frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_n}}\leq\sqrt[n]{x_1x_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}}.</math><ref>{{Cite book |last=Djukić |first=Dušan |title=The IMO compendium: a collection of problems suggested for the International Mathematical Olympiads, 1959-2009 |date=2011 |publisher=Springer |others=International mathematical olympiad |isbn=978-1-4419-9854-5 |series=Problem books in mathematics |location=New York |pages=7}}</ref>

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 multiplier]]s, 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#real_number_proof|Cauchy–Schwarz inequality on real numbers]], setting one vector to {{math|(1, 1, ...)}}:

:<math>\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,</math> hence <math>\left( \frac{\sum_{i=1}^n x_i}{n} \right)^2 \leq \frac{\sum_{i=1}^n x_i^2}{n}</math>. For positive <math>x_i</math> the square root of this gives the inequality.

===HM-GM inequality===
 
The reciprocal of the harmonic mean is the arithmetic mean of the reciprocals <math>1/x_1  , \dots, 1/x_n</math>, and it exceeds <math>1/\sqrt[n]{x_1 \dots x_n}</math> by the AM-GM inequality. <math>x_i > 0</math> implies the inequality:

:<math> \frac{n}{\frac{1}{x_1} + \dots + \frac{1}{x_n}} \leq \sqrt[n]{x_1\dots x_n}. </math><ref>{{Citation |last=Sedrakyan |first=Hayk |title=The HM-GM-AM-QM Inequalities |date=2018 |url=https://doi.org/10.1007/978-3-319-77836-5_3 |work=Algebraic Inequalities |pages=23 |editor-last=Sedrakyan |editor-first=Hayk |access-date=2023-11-26 |series=Problem Books in Mathematics |place=Cham |publisher=Springer International Publishing |language=en |doi=10.1007/978-3-319-77836-5_3 |isbn=978-3-319-77836-5 |last2=Sedrakyan |first2=Nairi |editor2-last=Sedrakyan |editor2-first=Nairi}}</ref>

== The ''n''&thinsp;=&thinsp;2 case ==
[[File:HM-GM-AM-QM inequality n=2 case visualization .jpg|thumb|The semi-circle used to visualize the inequalities]]
When ''n''&thinsp;=&thinsp;2, the inequalities become 
:<math>\frac {2 x_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}}</math> for all <math>x_1, x_2 > 0,</math> <ref>{{Citation |last=Sedrakyan |first=Hayk |title=The HM-GM-AM-QM Inequalities |date=2018 |url=https://doi.org/10.1007/978-3-319-77836-5_3 |work=Algebraic Inequalities |pages=21 |editor-last=Sedrakyan |editor-first=Hayk |access-date=2023-11-26 |series=Problem Books in Mathematics |place=Cham |publisher=Springer International Publishing |language=en |doi=10.1007/978-3-319-77836-5_3 |isbn=978-3-319-77836-5 |last2=Sedrakyan |first2=Nairi |editor2-last=Sedrakyan |editor2-first=Nairi}}</ref>
which can be visualized in a semi-circle whose diameter is [''AB''] and center&nbsp;''D''.

Suppose ''AC''&thinsp;=&thinsp;''x''<sub>1</sub> and ''BC''&thinsp;=&thinsp;''x''<sub>2</sub>. Construct perpendiculars to [''AB''] at ''D'' and ''C'' respectively. Join [''CE''] and [''DF''] and further construct a perpendicular [''CG''] to [''DF''] at ''G''. Then the length of ''GF'' can be calculated to be the harmonic mean, ''CF'' to be the geometric mean, ''DE'' to be the arithmetic mean, and ''CE'' to be the quadratic mean. The inequalities then follow easily by the [[Pythagorean theorem]].

[[File:Comparison_mean_values.svg|thumb|center|x300px|Comparison of harmonic, geometric, arithmetic, quadratic and other mean values of two positive real numbers <math>x_1</math> and <math>x_2</math>]]

== Tests ==
To infer the correct order, the four expressions can be evaluated with two positive numbers.

For <math>x_1=10</math> and <math>x_2=40</math> in particular, this results in <math>16 < 20 < 25 < 5 \sqrt{34} </math>.

== See also ==
*[[Pythagorean means#Inequalities among means|Inequalities among pythagorean means]]
*[[Generalized mean#Generalized mean inequality|Generalized mean inequality]]

==References==
{{reflist}}

== External links ==
*[https://www.maa.org/sites/default/files/gwan01200422828.pdf The HM-GM-AM-QM Inequalities]
*[http://www.lkozma.net/inequalities_cheat_sheet/ineq.pdf Useful inequalities cheat sheet] entry "means" on the right of page 1

[[Category:Inequalities]]