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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:

$$\prod _{{k=1}}^{n}(x_{k}+y_{k})^{{1/n}}\geq \prod _{{k=1}}^{n}x_{k}^{{1/n}}+\prod _{{k=1}}^{n}y_{k}^{{1/n}}$$

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.

Minkowski's inequality

References

http://eom.springer.de/M/m064060.htm