A simple proof of the generalized Schur inequality

For a given n×n matrix A with eigenvalues λ1,…,λn a new and a very simple proof of the generalized Schur inequality ∑i=1n|λi|p ⩽ ∑i,j=1n|aij|p, 1 ⩽ p < 2, is given. For the case p = 1 we derive this inequality in another and even simpler way. The cases of equality in the inequality above are discussed.