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?
Inequalities for a Sum over Elementary Symmetric Polynomials
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real-analysis
inequality
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1No better constants are possible. Taking $x_1=x_2=\cdots=x_n=t$ and sending $t \to 0^{+}$ shows that you can't increase $A$ above $1$, sending $t \to \infty$ shows that you can't lower $B$ below $n-1$. – 2012-06-20