马勒不等式在数学领域, 马勒不等式陈述说由两个无穷正项序列的对应项的和构成序列的几何均值大于或等于这两个无穷序列几何均值的和: ∏ 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}} 其中, 对任何的k, xk, yk > 0. 不等式以库尔特·马勒的名字命名. 证明 由均值不等式, 有: ∏ 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}},} 和 ∏ 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}}.} 因此, ∏ 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.} 整理后即得结论. 参阅 闵可夫斯基不等式参考文献 http://eom.springer.de/M/m064060.htm （页面存档备份，存于互联网档案馆）
