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:
∏ k = 1 n ( x k + y k ) 1 / n ≥ ∏ k = 1 n x k 1 / n + ∏ k = 1 n y k 1 / n {\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}}
when x k , y k > 0 {\displaystyle x_{k},\ y_{k}>0} for all k {\displaystyle k} .
Proof By the inequality of arithmetic and geometric means, we have:
∏ k = 1 n ( x k x k + y k ) 1 / n ≤ 1 n ∑ k = 1 n x k x k + y k , {\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
∏ k = 1 n ( y k x k + y k ) 1 / n ≤ 1 n ∑ k = 1 n y k x k + y k . {\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,
∏ k = 1 n ( x k x k + y k ) 1 / n + ∏ k = 1 n ( y k x k + y k ) 1 / n ≤ 1 n n = 1. {\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.
See also Minkowski inequality
References Minkowski inequality in the Encyclopedia of Mathematics