proof of arithmetic-geometric means inequality

A short geometrical proof can be given for the case $n=2$ of the arithmetic-geometric means inequality.

Let $a$ and $b$ be two non negative numbers. Draw the line $A B$ such that $A P$ has length $a$ , and $P B$ has length $b$ , as in the following picture, and draw a semicircle with diameter $A B$ . Let $O$ be the center of the circle.

Now raise perpendiculars $P Q$ and $O T$ to $A B$ . Notice that $O T$ is a radius, and so

$OT=\frac{AB}{2}=\frac{a+b}{2}$

Also notice that $PQ\leq OT$ for any point $P$ , and equality is obtained only when $P=O$ , that is, when $a=b$ .

Notice also that $P Q$ is a height over the hypotenuse on right triangle $\triangle AQB$ . We have then triangle similarities $\triangle AQB\sim\triangle APQ\sim\triangle QPB$ , and thus

$\frac{AP}{PQ}=\frac{PQ}{PB}$

which implies $PQ=\sqrt{AP\cdot PB}=\sqrt{ab}$ . Since $PQ\leq OT$ , we conclude

$\sqrt{ab}\leq\frac{a+b}{2}.$

This special case can also be proved using rearrangement inequality. Let $a, b$ non negative numbers, and assume $a\leq b$ . Let $x_{1}=\sqrt{a},x_{2}=\sqrt{b}$ , and then $x_{1}\leq x_{2}$ . Now suppose $y_{1}$ and $y_{2}$ are such that one of them is $x_{1}$ and the other is $x_{2}$ . Rearrangement inequality states that $x_{1}y_{1}+x_{2}y_{2}$ is maximum when $y_{1}\leq y_{2}$ and $x_{1}\leq x_{2}$ . So, we have

$x_{1}x_{2}+x_{2}x_{1}\leq x_{1}^{2}+x_{2}^{2}$

and substituting back $a, b$ gives

$2\sqrt{ab}\leq(\sqrt{a})^{2}+(\sqrt{b})^{2}=a+b$

where it follows the desired result.

One more proof can be given as follows. Let $x=\sqrt{a},y=\sqrt{b}$ . Then $(x-y)^{2}\geq 0$ , and equality holds only when $x=y$ . Then, $x^{2}-2xy+y^{2}\geq 0$ becomes

$x^{2}+y^{2}\geq 2xy$

and substituting back $a, b$ gives the desired result as in the previous proof.

Title	proof of arithmetic-geometric means inequality
Canonical name	ProofOfArithmeticgeometricMeansInequality
Date of creation	2013-03-22 14:49:14
Last modified on	2013-03-22 14:49:14
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	10
Author	mathcam (2727)
Entry type	Proof
Classification	msc 26D15