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
x1,x2,\ldots,xn
0< | n | |||||||||
|
\leq\sqrt[n]{x1x2 …
x | \leq\sqrt{ | ||||
|
| |||||||
n |
These inequalities often appear in mathematical competitions and have applications in many fields of science.
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 :
\left(
n | |
\sum | |
i=1 |
xi ⋅ 1\right)2\leq\left(
n | |
\sum | |
i=1 |
2 | |
x | |
i |
\right)\left(
n | |
\sum | |
i=1 |
12\right)=n
n | |
\sum | |
i=1 |
2, | |
x | |
i |
\left(
| ||||||||||
n |
\right)2\leq
| ||||||||||||||||
n |
xi
The reciprocal of the harmonic mean is the arithmetic mean of the reciprocals
1/x1,...,1/xn
1/\sqrt[n]{x1...xn}
xi>0
n | |||||||
|
\leq\sqrt[n]{x1...xn}.
When n = 2, the inequalities become
2 | |||||||
|
\leq\sqrt{x1x2}\leq
x1+x2 | \leq\sqrt{ | |
2 |
| |||||||
2 |
x1,x2>0,
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.
To infer the correct order, the four expressions can be evaluated with two positive numbers.
For
x1=10
x2=40
16<20<25<5\sqrt{34}