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, \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 | 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 | ||
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<center><math>\left(1 + \frac {x}{y}\right)^k + \left(1 + \frac {y}{x}\right)^k\geq 2^{k+1}</math></center> | <center><math>\left(1 + \frac {x}{y}\right)^k + \left(1 + \frac {y}{x}\right)^k\geq 2^{k+1}</math></center> | ||
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[[Category:Inequality]] | [[Category:Inequality]] | ||
[[Category:Definition]] | [[Category:Definition]] | ||
[[Category:Theorems]] | [[Category:Theorems]] |
Latest revision as of 11:12, 29 October 2016
See also: Hölder's inequality
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: