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:
\( \prod _{{k=1}}^{n}(x_{k}+y_{k})^{{1/n}}\geq \prod _{{k=1}}^{n}x_{k}^{{1/n}}+\prod _{{k=1}}^{n}y_{k}^{{1/n}} \)
when xk, yk > 0 for all k.
Proof
By the inequality of arithmetic and geometric means, we have:
\( \prod _{{k=1}}^{n}\left({x_{k} \over x_{k}+y_{k}}\right)^{{1/n}}\leq {1 \over n}\sum _{{k=1}}^{n}{x_{k} \over x_{k}+y_{k}}, \)
and
\( \prod _{{k=1}}^{n}\left({y_{k} \over x_{k}+y_{k}}\right)^{{1/n}}\leq {1 \over n}\sum _{{k=1}}^{n}{y_{k} \over x_{k}+y_{k}}. \)
Hence,
\( \prod _{{k=1}}^{n}\left({x_{k} \over x_{k}+y_{k}}\right)^{{1/n}}+\prod _{{k=1}}^{n}\left({y_{k} \over x_{k}+y_{k}}\right)^{{1/n}}\leq {1 \over n}n=1. \)
Clearing denominators then gives the desired result.
See also
Minkowski's inequality
References
http://eom.springer.de/M/m064060.htm
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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