To state the question and fix conventions I will introduce some notation from e.g. (Lin-Trudinger, Bull. Aust. Math. Soc. 1994, ``On some inequalities for elementary symmetric functions")

Given $\lambda_1, \dots, \lambda_n$ let

\begin{align*} \sigma_k(\lambda) = \sum_{1 \leq i_1 < i_2 < \dots < i_k \leq n} \lambda_{i_1} \lambda_{i_2} \dots \lambda_{i_k} \end{align*} We will always consider this function as restricted to the ``positive cone" \begin{align*} \Gamma_k = \left\{ \lambda \in \mathbb R^n\ |\ \sigma_j(\lambda) > 0 \mbox{ for all } 1 \leq j \leq k \right\} \end{align*}

Furthermore, let \begin{align*} \sigma_{k;i}(\lambda) = \sigma_k(\lambda)_{| \lambda_{i} = 0}, \end{align*} i.e. the function $\sigma_k(\lambda)$ after replacing $\lambda_i = 0$. To see some familiar facts with this notation, note the Newton inequalities \begin{align*} \sigma_k(\lambda) \sigma_{k-2}(\lambda) \leq \frac{(k-1)(n-k+1)}{k(n-k+2)} [\sigma_{k-1}(\lambda)]^2, \end{align*} as well as the Maclaurin inequality for $\lambda \in \Gamma_k$, $k \geq l \geq 1$. \begin{align*} \left[ \frac{1}{n \choose k} \sigma_k(\lambda)\right]^{\frac{1}{k}} \leq \left[ \frac{1}{n \choose l} \sigma_l(\lambda)\right]^{\frac{1}{l}} \end{align*} One can observe the further elementary identities \begin{align*} \sigma_k(\lambda) =&\ \sigma_{k;i}(\lambda) + \lambda_i \sigma_{k-1;i}(\lambda),\\ \sum_{i=1}^n \sigma_{k;i}(\lambda) =&\ (n-k) \sigma_k(\lambda). \end{align*}

Now to state the actual question. Fix $n = 2k$. Given $\lambda = (\lambda_1, \dots, \lambda_n)$, let \begin{align*} A_{\lambda} =&\ \frac{1}{n-2} \left[ \lambda - \frac{\sigma_1(\lambda)}{2(n-1)} (1,\dots,1) \right] \end{align*} I now assume I have a vector $\lambda$ such that $A_{\lambda} \in \Gamma_k$ (recall $k = \frac{n}{2}$). The inequality I seek is that for all $1 \leq i \leq n$, \begin{align*} (n-1) \sigma_k(A_{\lambda}) \leq \lambda_i \sigma_{k-1;i}(A_{\lambda}). \end{align*} A few remarks. First, if helpful it can be expressed purely in terms of the vector $A_{\lambda}$, by using the formula $\lambda = (n-2) A_{\lambda} + \sigma_1(A_{\lambda}) (1,\dots,1)$. Second, it is known that the condition $A_{\lambda} \in \Gamma_k$ implies that $\lambda_i \geq 0$ for all $i$. Third, I have added the tag, ``differential geometry'' only because this question originally comes from a geometric problem, where $\lambda$ represents eigenvalues of the Ricci tensor and $A_{\lambda}$ is the Schouten tensor. Lastly, it is a fairly straightforward matter to verify that the inequality is true for $n=2,4$. Any help is greatly appreciated!