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.)
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Latest revision as of 18:42, 1 March 2021
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.
If , , then and .
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
- Prove that, for positive reals , the following inequality holds: