# Difference between revisions of "Aczel's Inequality"

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be <math>m</math> sequences of positive real numbers such that <math>a_{i1}^{ p_i} - a_{i2}^{ p_i} - \dots - a_{in}^{ p_i} > 0</math> for <math>i=1,\dots,m </math>. Then | be <math>m</math> sequences of positive real numbers such that <math>a_{i1}^{ p_i} - a_{i2}^{ p_i} - \dots - a_{in}^{ p_i} > 0</math> for <math>i=1,\dots,m </math>. Then | ||

− | <center><math> \prod_{i=1}^m | + | <center><math> \prod_{i=1}^m a_{i1} - \prod_{i=1}^m a_{i2} -\dots- \prod_{i=1}^m a_{in} \ge\prod_{i=1}^m |

− | + | (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. | ||

− | |||

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

## Revision as of 12:58, 11 March 2011

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

## Contents

## 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.

## 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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