Difference between revisions of "Hölder's Inequality"
m (I'm trying to see if removing the category "Theorems" will fix the issue described here: https://artofproblemsolving.com/community/c10h2471130_formatting_error_on_aops_wiki) |
m (Adding back the theorems category; it did not fix the issue sadly.) |
||
Line 42: | Line 42: | ||
[[Category:Inequality]] | [[Category:Inequality]] | ||
[[Category:Definition]] | [[Category:Definition]] | ||
+ | [[Category:Theorems]] |
Revision as of 17:42, 1 March 2021
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:
