Difference between revisions of "Hölder's Inequality"
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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 CauchySchwarz 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 AMGM. We will disregard sequences for which one of the terms is zero, as the terms of these sequences do not contribute to the lefthand side of the desired inequality but may contribute to the righthand side.
For integers , let us define Evidently, . Then for all integers , by weighted AMGM, 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: