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.
For any real numbers and ,
with equality when there exist constants not both zero such that for all , .
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. Note that this is not actually a proof; rather, this result is used to establish that we can define the angles between two vectors in this way. Indeed, the proof that structures such as vectors exist in using linear algebra requires this very inequality, and so this would be circular logic.
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
with equality if and only if there exist constants not both zero such that .
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.
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.
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 ,
- Consider the function , where is a positive integer. Show that . (Source)
- 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)
- is a point inside a given triangle . are the feet of the perpendiculars from to the lines , respectively. Find all for which
- 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.