Let $(x_1,\ldots,x_n) \in (\mathbb{R}_+)^n$. How can I prove that $$2\max(x_1,\ldots,x_n)\left(\frac{1}{n}\sum_1^n x_k - \prod_1^n x_k^{1/n}\right) \ge \frac{1}{n}\sum_1^n\left(x_k - \prod_1^n x_j^{1/n}\right)^2$$
Note : all the sums and products start at $k=1$ (but i didn't manage to write with LaTeX)