weighted arithmetic-geometric mean inequality

Generalized form of the AM-GM inequality
\[ \sum_i \alpha_i v_i \ge \prod_i v_i^{\alpha_i} \] where \(\mathbf{v} \succ 0\) and \(\mathbf{\alpha} \succeq 0,\; \mathbf{1}^T\mathbf{\alpha} = 1\).