# 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 $AB$ such that $AP$ has length $a$, and $PB$ has length $b$, as in the following picture, and draw a semicircle with diameter^{} $AB$. Let $O$ be the center of the circle.

Now raise perpendiculars^{} $PQ$ and $OT$ to $AB$. Notice that $OT$ is a radius, and so

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

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

Notice also that $PQ$ is a height over the hypotenuse^{} on right triangle^{} $\mathrm{\u25b3}AQB$. We have then triangle similarities^{} $\mathrm{\u25b3}AQB\sim \mathrm{\u25b3}APQ\sim \mathrm{\u25b3}QPB$, and thus

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

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

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

This special case can also be proved using rearrangement inequality. Let $a,b$ non negative numbers, and assume $a\le b$. Let ${x}_{1}=\sqrt{a},{x}_{2}=\sqrt{b}$, and then ${x}_{1}\le {x}_{2}$. 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 ${y}_{1}\le {y}_{2}$ and ${x}_{1}\le {x}_{2}$. So, we have

$${x}_{1}{x}_{2}+{x}_{2}{x}_{1}\le {x}_{1}^{2}+{x}_{2}^{2}$$ |

and substituting back $a,b$ gives

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

where it follows the desired result.

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

$${x}^{2}+{y}^{2}\ge 2xy$$ |

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 |