Bound of weighted trace of a self-adjoint matrix

Question

Let $A\in M_n(\mathbb{C})$ be self-adjoint matrix with eigenvalues $\lambda_1\geq\cdots\geq\lambda_n$. Suppose $\{e_j\}_{j=1}^n$ be an orthonormal basis of $\mathbb{C}^n$ and $c_1\geq c_2\geq\cdots \geq c_n$. Show that $$\sum_{j=1}^nc_j\lambda_{n-j+1}\leq\sum\limits_{j=1}^n c_j\langle Ae_j,e_j\rangle\leq\sum\limits_{j=1}^nc_j\lambda_j.$$