Mahler's Inequality

Theorem
The geometric mean of the termwise sum of two finite sequences of positive real numbers is never less than the sum of their two separate geometric means:


 * $\displaystyle \prod_{k \mathop = 1}^n \left({x_k + y_k}\right)^{1/n} \ge \prod_{k \mathop = 1}^n x_k^{1/n} + \prod_{k \mathop = 1}^n y_k^{1/n}$

where $x_k, y_k > 0$ for all $k$.

Proof
This leads to: