Let $\bar x = (x_1, x_2, \dots, x_n)$ and $\bar y = (y_1, y_2, \dots, y_n)$ be non-negative vectors in $\mathbb R^n$, and $\bar z = \bar x + \bar y$. For $1 \leq k \leq n$, define the $k$-th symmetric polynomial of $\bar x$ as
$$\sigma_k(\bar x) = \sum_{1 \leq i_1 < i_2 \dots < i_k \leq n} x_{i_1}x_{i_2}\dots x_{i_k},$$
and similarly define $\sigma_k(\bar y)$ and $\sigma_k(\bar z)$.
Prove that the following inequality holds:
$$\sigma_k(\bar x)^{\frac{1}{k}} + \sigma_k(\bar y)^{\frac{1}{k}} \leq \sigma_k(\bar z)^{\frac{1}{k}}$$
