Cauchy-Schwarz Inequality
In algebra, the Cauchy-Schwarz Inequality, also known as the Cauchy-Bunyakovsky-Schwarz Inequality or informally as Cauchy-Schwarz, is a ubiquitous inequality with many formulations. Its most general form is written in abstract vector spaces, but in contest math, its applications are limited to elementary algebraic and linear algebra.
Its elementary algebraic formulation is often referred to as Cauchy's Inequality and states the following: for any list of reals and
,
with equality if and only if there exists a constant
such that
for all
, or if all the entries in one of the lists are zero. Along with the AM-GM Inequality, Cauchy-Schwarz forms the foundation for inequality problems in intermediate and olympiad competitions. It is particularly crucial in proof-based inequalities.
Its linear algebra forulation states the following: for any vectors and
in
,
with equality if and only if there exists a scalar
such that
, or of one of the vectors is zero.
Contents
Elementary Form
For any real numbers and
,
with equality when there exists a nonzero constant
such that for all
,
.
To easily memorize this formula, we use the simpler, more compact notation:
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.