Throughout this part, $n$ denotes an integer greater than or equal to 3, and $p = [(n+1)/2]$ is the integer part of $(n+1)/2$. We admit the following result: if $A$ and $B$ are two matrices of $\mathcal{S}_n(\mathbb{R})$ whose respective eigenvalues (with possible repetitions) are $\alpha_1 \geqslant \ldots \geqslant \alpha_n$ and $\beta_1 \geqslant \ldots \geqslant \beta_n$ then $$\sum_{i=1}^{n} \alpha_i \beta_{n+1-i} \leqslant \operatorname{tr}(AB) \leqslant \sum_{i=1}^{n} \alpha_i \beta_i$$ Let $B = (b_{i,j})_{1 \leqslant i,j \leqslant n}$ be the matrix of $\mathcal{M}_n(\mathbb{R})$ defined by $$b_{1,2p-1} = 1 \quad b_{2p-1,1} = 1 \quad b_{p,p} = -2$$ all other coefficients of $B$ being zero. Let $a = (a_0, \ldots, a_{2n-2})$ be an element of $\mathbb{R}^{2n-1}$ and $M = H(a)$. We denote $\operatorname{Spo}(M) = (\lambda_1, \ldots, \lambda_n)$. Establish that $$\lambda_1 - \lambda_{n-1} - 2\lambda_n \geqslant 0 \quad \text{and} \quad 2\lambda_1 + \lambda_2 - \lambda_n \geqslant 0 \tag{III.3}$$
Throughout this part, $n$ denotes an integer greater than or equal to 3, and $p = [(n+1)/2]$ is the integer part of $(n+1)/2$.
We admit the following result: if $A$ and $B$ are two matrices of $\mathcal{S}_n(\mathbb{R})$ whose respective eigenvalues (with possible repetitions) are $\alpha_1 \geqslant \ldots \geqslant \alpha_n$ and $\beta_1 \geqslant \ldots \geqslant \beta_n$ then
$$\sum_{i=1}^{n} \alpha_i \beta_{n+1-i} \leqslant \operatorname{tr}(AB) \leqslant \sum_{i=1}^{n} \alpha_i \beta_i$$
Let $B = (b_{i,j})_{1 \leqslant i,j \leqslant n}$ be the matrix of $\mathcal{M}_n(\mathbb{R})$ defined by
$$b_{1,2p-1} = 1 \quad b_{2p-1,1} = 1 \quad b_{p,p} = -2$$
all other coefficients of $B$ being zero.
Let $a = (a_0, \ldots, a_{2n-2})$ be an element of $\mathbb{R}^{2n-1}$ and $M = H(a)$. We denote $\operatorname{Spo}(M) = (\lambda_1, \ldots, \lambda_n)$.
Establish that
$$\lambda_1 - \lambda_{n-1} - 2\lambda_n \geqslant 0 \quad \text{and} \quad 2\lambda_1 + \lambda_2 - \lambda_n \geqslant 0 \tag{III.3}$$