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:

\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, x_k, y_k > 0.

不等式以库尔特·马勒的名字命名.

证明

\prod_{k = 1}^{n} {(\frac{x_{k}}{x_{k} + y_{k}})}^{1 / n} \leq \frac{1}{n} \sum_{k = 1}^{n} \frac{x_{k}}{x_{k} + y_{k}},

和

\prod_{k = 1}^{n} {(\frac{y_{k}}{x_{k} + y_{k}})}^{1 / n} \leq \frac{1}{n} \sum_{k = 1}^{n} \frac{y_{k}}{x_{k} + y_{k}} .

因此,

\prod_{k = 1}^{n} {(\frac{x_{k}}{x_{k} + y_{k}})}^{1 / n} + \prod_{k = 1}^{n} {(\frac{y_{k}}{x_{k} + y_{k}})}^{1 / n} \leq \frac{1}{n} n = 1 .

整理后即得结论.