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
[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.
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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