# Difference between revisions of "Cauchy-Schwarz Inequality"

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with equality when <math>\mathbf{a} </math> and <math> \mathbf{b} </math> may be scaled to each other, as desired. | with equality when <math>\mathbf{a} </math> and <math> \mathbf{b} </math> may be scaled to each other, as desired. | ||

+ | |||

+ | === Proof 3 === | ||

+ | |||

+ | Consider <math>a-\lambda b</math> for some scalar <math>\lambda</math>. Then: | ||

+ | <math>0\le||a-\lambda b||^2</math> (by the Trivial Inequality) | ||

+ | <math>=\langle a-\lambda b,a-\lambda b\rangle</math> | ||

+ | <math>=\langle a,a\rangle-2\lambda\langle a,b\rangle+\lambda^2\langle y,y\rangle</math> | ||

+ | <math>=||a||^2-2\lambda\langle a,b\rangle+\lambda^2||b||^2</math> | ||

+ | Now, let <math>\lambda=\frac{\langle a,b\rangle}{||b||^2}</math>. Then, we have: | ||

+ | <math>0\le||a||^2-\frac{\langle a,b\rangle|^2}{||b||^2}</math> | ||

+ | <math>\implies\langle a,b\rangle|^2\le||a||^2||b||^2=\langle a,a\rangle\cdot\langle b,b\rangle</math>. <math>\square</math> | ||

=== Examples === | === Examples === |

## Revision as of 14:55, 17 January 2021

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.

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

## Contents

## Elementary Form

For any real numbers and , with equality when there exists a nonzero constant 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 with equality if and only if, for some ordering of the pairs , some exists such that for and for , and If we restrict that and for all , then it's clear that for to be or for all , then and , so is equivalent to (When this is not an integer, the maximum occurs when is either the ceiling or floor of the right-hand side.) In the special case that is constant for all , we have and , so here must be .

### 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. For equality to occur, we must have or for all . If, without loss of generality, for and for for some , then for the AM-GM to reach equality we must have (assume since is trivial)

## 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.

### Proof 3

Consider for some scalar . Then: (by the Trivial Inequality) Now, let . Then, we have: .

### 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.