We denote $\mathcal{B}_{n}(\mathbb{R})$ the set of doubly stochastic matrices in $\mathcal{M}_{n}(\mathbb{R})$, that is the set of matrices $M = \left(m_{i,j}\right)_{1 \leqslant i,j \leqslant n}$ whose coefficients are all non-negative and such that $\sum_{j=1}^{n} m_{i,j} = \sum_{j=1}^{n} m_{j,i} = 1$ for every $i \in \llbracket 1, n \rrbracket$. We denote $f : \left|\, \begin{array}{ccc} \mathcal{M}_{n}(\mathbb{R}) & \rightarrow & \mathbb{R} \\ M & \mapsto & \sum_{1 \leqslant i,j \leqslant n} m_{i,j}\left(\lambda_{i}(A) - \lambda_{j}(B)\right)^{2}. \end{array}\right.$ Let $(i,j,k) \in \llbracket 1,n \rrbracket^{3}$ such that $j \geqslant i$ and $k \geqslant i$. Show that, for $M \in \mathcal{M}_{n}(\mathbb{R})$ and for $x \in \mathbb{R}^{+}$, $$f\left(M + xE_{ii} + xE_{jk} - xE_{ik} - xE_{ji}\right) - f(M) = 2x\left(\lambda_{i}(A) - \lambda_{j}(A)\right)\left(\lambda_{k}(B) - \lambda_{i}(B)\right) \leqslant 0$$
We denote $\mathcal{B}_{n}(\mathbb{R})$ the set of doubly stochastic matrices in $\mathcal{M}_{n}(\mathbb{R})$, that is the set of matrices $M = \left(m_{i,j}\right)_{1 \leqslant i,j \leqslant n}$ whose coefficients are all non-negative and such that $\sum_{j=1}^{n} m_{i,j} = \sum_{j=1}^{n} m_{j,i} = 1$ for every $i \in \llbracket 1, n \rrbracket$.
We denote $f : \left|\, \begin{array}{ccc} \mathcal{M}_{n}(\mathbb{R}) & \rightarrow & \mathbb{R} \\ M & \mapsto & \sum_{1 \leqslant i,j \leqslant n} m_{i,j}\left(\lambda_{i}(A) - \lambda_{j}(B)\right)^{2}. \end{array}\right.$
Let $(i,j,k) \in \llbracket 1,n \rrbracket^{3}$ such that $j \geqslant i$ and $k \geqslant i$. Show that, for $M \in \mathcal{M}_{n}(\mathbb{R})$ and for $x \in \mathbb{R}^{+}$,
$$f\left(M + xE_{ii} + xE_{jk} - xE_{ik} - xE_{ji}\right) - f(M) = 2x\left(\lambda_{i}(A) - \lambda_{j}(A)\right)\left(\lambda_{k}(B) - \lambda_{i}(B)\right) \leqslant 0$$