Difference between revisions of "Aczel's Inequality"
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(a_{i1}^{ p_i} - a_{i2}^{ p_i} - \dots - a_{in}^{ p_i})^\frac 1{ p_i}</math></center> | (a_{i1}^{ p_i} - a_{i2}^{ p_i} - \dots - a_{in}^{ p_i})^\frac 1{ p_i}</math></center> | ||
with equality if and only if all the sequences are proportional. | with equality if and only if all the sequences are proportional. | ||
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+ | ==Examples== | ||
+ | |||
+ | '''Olympiad''' | ||
+ | Suppose <math>a_1, a_2, \ldots, a_n</math> and <math>b_1, b_2, \ldots, b_n</math> are real numbers such that <cmath> (a_1 ^ 2 + a_2 ^ 2 + \cdots + a_n ^ 2 -1)(b_1 ^ 2 + b_2 ^ 2 + \cdots + b_n ^ 2 - 1) > (a_1 b_1 + a_2 b_2 + \cdots + a_n b_n - 1)^2. </cmath> Prove that <math>a_1 ^ 2 + a_2 ^ 2 + \cdots + a_n ^ 2 > 1</math> and <math>b_1 ^ 2 + b_2 ^ 2 + \cdots + b_n ^ 2 > 1</math>. | ||
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== References == | == References == |
Revision as of 14:24, 28 June 2014
Aczél's Inequality states that if or , then
Contents
[hide]Proof
Consider the function .
We have , and from we get . Therefore, must have at least one root, .
General Form
Let such that and let
be sequences of positive real numbers such that for . Then
with equality if and only if all the sequences are proportional.
Examples
Olympiad Suppose and are real numbers such that Prove that and .
References
- Mascioni, Vania, A note on Aczél-type inequalities, JIPAM volume 3 (2002), issue 5, article 69.
- Popoviciu, T., Sur quelques inégalités, Gaz. Mat. Fiz. Ser. A, 11 (64) (1959) 451–461
See also
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