Define $f(x_1, \dots, x_n) = \sum_{l = 2}(l - 1) \sigma_{l}(x_1, \dots, x_n)$, where $\sigma_{l}$ is the $l^{\text{th}}$-elementary symmetric polynomial and $(x_1, \dots, x_n)$ is non-negative. Beyond the obvious bounds \begin{align} \sum_{l = 2}^{n} \sigma_{l}(x_1, \dots, x_n) \leq f(x_1, \dots, x_n) \leq (n-1) \sum_{l = 2}^{n} \sigma_{l}(x_1, \dots, x_n), \end{align} are there sharper inequalities on the sums which can be used to yield something of the sort: \begin{align} A \sum_{l = 2}^{n} \sigma_{l}(x_1, \dots, x_n) \leq f(x_1, \dots, x_n) \leq B \sum_{l = 2}^{n} \sigma_{l}(x_1, \dots, x_n), \end{align} for some constants (or perhaps functions) $A > 1$ and $0 < B < n - 1$? If not, what other inequalities are suitable for this problem?