QM-AM-GM-HM inequalities: Difference between revisions

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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, geometric mean, arithmetic mean, and quadratic mean (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]

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

\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.

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

The semi-circle used to visualize the inequalities

When n = 2, the inequalities become

{\frac {2}{{\frac {1}{x_{1}}}+{\frac {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 [AB] and center D.

Suppose AC = x₁ and BC = x₂. 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.

Comparison of harmonic, geometric, arithmetic, quadratic and other mean values of two positive real numbers $x_{1}$ and $x_{2}$

Tests

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

For $x_{1}=10$ and $x_{2}=40$ in particular, this results in $16<20<25<5{\sqrt {34}}$ .

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
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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@@ Line 1: / Line 1: @@
+{{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>
-{{short description|mathematical relationships}}
-In [[mathematics]], the '''HM-GM-AM-QM inequalities''' state the relationship between the [[harmonic mean]], [[geometric mean]], [[arithmetic mean]], and [[Root mean square|quadratic mean]] (aka root mean square, RMS). Suppose that <math>x_1, x_2, \ldots, x_n</math> are positive [[real number]]s. Then
-: <math>0<\frac{n}{1/x_1+1/x_2+\cdots+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>
 These inequalities often appear in mathematical competitions and have applications in many fields of science.
@@ Line 15: / Line 14: @@
 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 u_i \cdot 1 \right)^2 \leq \left( \sum_{i=1}^n u_i^2 \right) \left( \sum_{i=1}^n 1^2 \right) = n \,\sum_{i=1}^n u_i^2,</math> hence <math>\left( \frac{\sum_{i=1}^n u_i}{n} \right)^2 \leq \frac{\sum_{i=1}^n u_i^2}{n}</math>. For positive <math>u_i</math> the square root of this gives the inequality.
+:<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=2n case visualization .jpg|thumb|The semi-circle used to visualize the inequalities]]
+[[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
-When ''n''&thinsp;=&thinsp;2, the inequalities become <math>\frac 2 {\frac{1}{x_1}+\frac{1}{x_2}} \leq \sqrt{x_1x_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> which can be visualized in a semi-circle whose diameter is [''AB''] and center&nbsp;''D''.
+:<math>\frac 2 {\frac{1}{x_1}+\frac{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 Data]
+*[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
-*[https://sites.math.washington.edu/~dumitriu/Inequalities]
 [[Category:Inequalities]]