Cauchy-Schwarz Inequality
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.
Contents
[hide]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 immediately from
.
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)
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.