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In [[mathematics]], '''Mahler's inequality''', named after [[Kurt Mahler]], states that the [[geometric mean]] of the term-by-term sum of two finite sequences of positive numbers is greater than or equal to the sum of their two separate geometric means:
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:<math>\prod_{k=1}^n (x_k + y_k)^{1/n} \ge \prod_{k=1}^n x_k^{1/n} + \prod_{k=1}^n y_k^{1/n}</math>
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when ''x''<sub>''k''</sub>, ''y''<sub>''k''</sub> > 0 for all ''k''.
 
== Proof ==
By the [[inequality of arithmetic and geometric means]], we have:
 
:<math>\prod_{k=1}^n \left({x_k \over x_k + y_k}\right)^{1/n} \le {1 \over n} \sum_{k=1}^n {x_k \over x_k + y_k},</math>
 
and
 
: <math>\prod_{k=1}^n \left({y_k \over x_k + y_k}\right)^{1/n} \le {1 \over n} \sum_{k=1}^n {y_k \over x_k + y_k}.</math>
 
Hence,
 
:<math>\prod_{k=1}^n \left({x_k \over x_k + y_k}\right)^{1/n}  + \prod_{k=1}^n \left({y_k \over x_k + y_k}\right)^{1/n} \le {1 \over n} n = 1.</math>
 
Clearing denominators then gives the desired result.  
 
== See also ==
*[[Minkowski's inequality]]
 
== References ==
*http://eom.springer.de/M/m064060.htm
 
[[Category:Inequalities]]
[[Category:Articles containing proofs]]
 
 
 
{{mathanalysis-stub}}

Latest revision as of 00:19, 4 September 2014

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