Difference between revisions of "H�lder's Inequality"
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== Elementary Form == | == Elementary Form == | ||
− | If <math>a_1, a_2, \dotsc, a_n, b_1, b_2, \dotsc, b_n, z_1, z_2, \dotsc, z_n</math> are [[nonnegative]] [[real number]]s and <math>\lambda_a, \lambda_b, \dotsc, \lambda_z</math> are nonnegative reals with sum of 1, then | + | If <math>a_1, a_2, \dotsc, a_n, b_1, b_2, \dotsc, b_n, \dotsc, z_1, z_2, \dotsc, z_n</math> are [[nonnegative]] [[real number]]s and <math>\lambda_a, \lambda_b, \dotsc, \lambda_z</math> are nonnegative reals with sum of 1, then |
<cmath> \begin{align*} | <cmath> \begin{align*} | ||
a_1^{\lambda_a}b_1^{\lambda_b} \dotsm z_1^{\lambda_z} + \dotsb &+ a_n^{\lambda_a} b_n^{\lambda_b} \dotsm z_n^{\lambda_z} \ | a_1^{\lambda_a}b_1^{\lambda_b} \dotsm z_1^{\lambda_z} + \dotsb &+ a_n^{\lambda_a} b_n^{\lambda_b} \dotsm z_n^{\lambda_z} \ |
Revision as of 21:47, 26 December 2008
Elementary Form
If are nonnegative real numbers and
are nonnegative reals with sum of 1, then
Note that with two sequences
and
, and
, this is the elementary form of the Cauchy-Schwarz Inequality.
We can state the inequality more concisely thus: Let be several sequences of nonnegative reals, and let
be a sequence of nonnegative reals such that
. Then
Proof of Elementary Form
We will use weighted AM-GM. We will disregard sequences for which one of the terms is zero, as the terms of these sequences do not contribute to the left-hand side of the desired inequality but may contribute to the right-hand side.
For integers , let us define
Evidently,
. Then for all integers
, by weighted AM-GM,
Hence
But from our choice of
, for all integers
,
Therefore
since the sum of the
is one. Hence in summary,
as desired. Equality holds when
for all integers
, i.e., when all the sequences
are proportional.
Statement
If ,
,
then
and
.
Proof
If then
a.e. and there is nothing to prove. Case
is similar. On the other hand, we may assume that
for all
. Let
. Young's Inequality gives us
These functions are measurable, so by integrating we get
Examples
- Prove that, for positive reals
, the following inequality holds:
