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:
![{\displaystyle \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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/932b88e1196eb4066bb1409ffdea26e0806674fc)
when xk, yk > 0 for all k.
Proof
By the inequality of arithmetic and geometric means, we have:
![{\displaystyle \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}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac9c12b30414f42c57dfd4e5b561f77af6e997fd)
and
![{\displaystyle \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}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8ad86f7a505bce3123c4cbdf38476cb19e57874)
Hence,
![{\displaystyle \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.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/02d190f2eef5fb4cc9f8b86086c258754ea5fe21)
Clearing denominators then gives the desired result.
See also
References
- Minkowski inequality in the Encyclopedia of Mathematics