# Difference between revisions of "Cauchy-Schwarz Inequality"

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For any real numbers <math> a_1, \ldots, a_n </math> and <math> b_1, \ldots, b_n </math>, | For any real numbers <math> a_1, \ldots, a_n </math> and <math> b_1, \ldots, b_n </math>, | ||

<cmath> | <cmath> | ||

− | \ | + | \left( \sum_{i=1}^{n}a_ib_i \right)^2 \le \left(\sum_{i=1}^{n}a_i^2 \right) \left(\sum_{i=1}^{n}b_i^2 \right), |

</cmath> | </cmath> | ||

with equality when there exist constants <math>\mu, \lambda </math> not both zero such that for all <math> 1 \le i \le n </math>, <math>\mu a_i = \lambda b_i </math>. | with equality when there exist constants <math>\mu, \lambda </math> not both zero such that for all <math> 1 \le i \le n </math>, <math>\mu a_i = \lambda b_i </math>. | ||

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Let <math> a_1, \ldots, a_n </math> and <math> b_1, \ldots, b_n </math> be [[complex numbers]]. Then | Let <math> a_1, \ldots, a_n </math> and <math> b_1, \ldots, b_n </math> be [[complex numbers]]. Then | ||

<cmath> | <cmath> | ||

− | \ | + | \left| \sum_{i=1}^na_ib_i \right|^2 \le \left(\sum_{i=1}^{n}|a_i^2| \right) \left( \sum_{i=1}^n |b_i^2| \right) . |

</cmath> | </cmath> | ||

This appears to be more powerful, but it follows from | This appears to be more powerful, but it follows from | ||

<cmath> | <cmath> | ||

− | \ | + | \left| \sum_{i=1}^n a_ib_i \right| ^2 \le \left( \sum_{i=1}^n |a_i| \cdot |b_i| \right)^2 \le \left(\sum_{i=1}^n |a_i^2| \right) \left( \sum_{i=1}^n |b_i^2| \right). |

</cmath> | </cmath> | ||

+ | |||

+ | == Upper Bound on (Σa)(Σb) == | ||

+ | |||

+ | Let <math>a_1, a_2, \ldots, a_n</math> and <math>b_1, b_2, \ldots, b_n</math> be two sequences of positive real numbers with | ||

+ | <cmath> | ||

+ | m \le \frac{a_i}{b_i} \le M | ||

+ | </cmath> | ||

+ | for <math>1 \le i \le n</math>. Then | ||

+ | <cmath> | ||

+ | \left(\sum_{i=1}^{n}a_i^2 \right) \left(\sum_{i=1}^{n}b_i^2 \right) \le \frac{(M+m)^2}{4Mm} \left( \sum_{i=1}^{n}a_ib_i \right)^2. | ||

+ | </cmath> | ||

+ | |||

+ | === Proof === | ||

+ | |||

+ | Note that for all <math>i</math>, we have | ||

+ | <cmath> | ||

+ | 0 \le \left(\frac{a_i}{b_i}-m\right)\left(M-\frac{a_i}{b_i}\right) = \frac{1}{b_i^2}(a_ib_iM-a_i^2-b_i^2Mm+a_ib_im) | ||

+ | </cmath> | ||

+ | or | ||

+ | <cmath> | ||

+ | (M+m)a_ib_i \ge a_i^2+(Mm)b_i^2, | ||

+ | </cmath> | ||

+ | with equality if and only if <math>a_i=mb_i</math> or <math>a_i=Mb_i</math>. Summing up these inequalities over <math>1 \le i \le n</math>, we obtain from AM-GM that | ||

+ | <cmath> | ||

+ | \begin{align*} | ||

+ | (M+m)\sum_{i=1}^{n}a_ib_i &\ge \sum_{i=1}^{n}a_i^2 + (Mm)\sum_{i=1}^{n}b_i^2\\ | ||

+ | &\ge 2\sqrt{(Mm) \left(\sum_{i=1}^{n}a_i^2 \right) \left(\sum_{i=1}^{n}b_i^2 \right)}, | ||

+ | \end{align*} | ||

+ | </cmath> | ||

+ | and squaring gives us the desired bound. | ||

== General Form == | == General Form == |

## Revision as of 00:35, 4 July 2010

The **Cauchy-Schwarz Inequality** (which is known by other names, including **Cauchy's Inequality**, **Schwarz's Inequality**, and the **Cauchy-Bunyakovsky-Schwarz Inequality**) is a well-known inequality with many elegant applications. It has an elementary form, a complex form, and a general form.

Augustin Louis Cauchy wrote the first paper about the elementary form in 1821. The general form was discovered by Viktor Bunyakovsky in 1849 and independently by Hermann Schwarz in 1888.

## Contents

## Elementary Form

For any real numbers and , with equality when there exist constants not both zero such that for all , .

### Discussion

Consider the vectors and . If is the angle formed by and , then the left-hand side of the inequality is equal to the square of the dot product of and , or . The right hand side of the inequality is equal to . The inequality then follows from , with equality when one of is a multiple of the other, as desired.

### Complex Form

The inequality sometimes appears in the following form.

Let and be complex numbers. Then This appears to be more powerful, but it follows from

## Upper Bound on (Σa)(Σb)

Let and be two sequences of positive real numbers with for . Then

### Proof

Note that for all , we have or with equality if and only if or . Summing up these inequalities over , we obtain from AM-GM that and squaring gives us the desired bound.

## General Form

Let be a vector space, and let be an inner product. Then for any , with equality if and only if there exist constants not both zero such that .

### Proof 1

Consider the polynomial of This must always be greater than or equal to zero, so it must have a non-positive discriminant, i.e., must be less than or equal to , with equality when or when there exists some scalar such that , as desired.

### Proof 2

We consider Since this is always greater than or equal to zero, we have Now, if either or is equal to , then . Otherwise, we may normalize so that , and we have with equality when and may be scaled to each other, as desired.

### Examples

The elementary form of the Cauchy-Schwarz inequality is a special case of the general form, as is the **Cauchy-Schwarz Inequality for Integrals**: for integrable functions ,
with equality when there exist constants not both equal to zero such that for ,

## Problems

### Introductory

- Consider the function , where is a positive integer. Show that . (Source)
- (APMO 1991 #3) Let , , , , , , , be positive real numbers such that . Show that

### Intermediate

- Let be a triangle such that

where and denote its semiperimeter and inradius, respectively. Prove that triangle is similar to a triangle whose side lengths are all positive integers with no common divisor and determine those integers. (Source)

### Olympiad

- is a point inside a given triangle . are the feet of the perpendiculars from to the lines , respectively. Find all for which

is least.

(Source)

## Other Resources

### Books

- The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities by J. Michael Steele.
- Problem Solving Strategies by Arthur Engel contains significant material on inequalities.