# Difference between revisions of "Aczel's Inequality"

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− | 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>. | + | 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>. (USA TST 2004) |

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== References == | == References == |

## Revision as of 10:50, 27 February 2020

**Aczél's Inequality** states that if or , then

## 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 . (USA TST 2004)

## 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

*This article is a stub. Help us out by expanding it.*