Mahler's Inequality

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


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

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

Proof
From Arithmetic Mean Never Less than Geometric Mean, we have:

$$ $$ $$ $$ $$ $$

This leads to:

$$ $$ $$ $$