Mahler's inequality

By the inequality of arithmetic and geometric means, we have:

${\displaystyle \prod _{k=1}^{n}\left({x_{k} \over x_{k}+y_{k}}\right)^{1/n}\leq {1 \over n}\sum _{k=1}^{n}{x_{k} \over x_{k}+y_{k}},}$

and

${\displaystyle \prod _{k=1}^{n}\left({y_{k} \over x_{k}+y_{k}}\right)^{1/n}\leq {1 \over n}\sum _{k=1}^{n}{y_{k} \over x_{k}+y_{k}}.}$

Hence,

${\displaystyle \prod _{k=1}^{n}\left({x_{k} \over x_{k}+y_{k}}\right)^{1/n}+\prod _{k=1}^{n}\left({y_{k} \over x_{k}+y_{k}}\right)^{1/n}\leq {1 \over n}n=1.}$

Clearing denominators then gives the desired result.

• Minkowski inequality

• Minkowski inequality in the Encyclopedia of Mathematics