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Revision as of 18:53, 18 December 2007
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
Elementary Form
For any real numbers and
,
,
with equality when there exist constants not both zero such that for all
,
.
Proof
There are several proofs; we will present an elegant one that does not generalize.
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.
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.