Some results on the Bergstrom and Minkowski inequalities
The main results are the following ones: 1. Suppose \(A\) and \(B\) are \(n \times n\) positive semidefinite matrices partitioned as \(P = (P_{ij})\), \(i,j = 1,2\), for \(P = A\) and \(P = B\), where \(P_{11}\) has order \(k\) and \(\text{det} P_{22} \neq 0\). \(P/P_{22} = P_{11} - P_{12} P^{-1}_{22} P_{21}\) denotes the Schur complement. Then: \[ A/A_{22} + B/B_{22} \leq (A + B)/(A_{22} + B_{22}). \] 2. Suppose \(A\) and \(B\) are \(M\)-matrices of order \(n\); let \(S \subseteq \{1,2, \dots, n\}\) such that \(|S |= 2\), and let \(i \in S\). Let \(X (P; S; i)\) be the matrix obtained from the \(n \times n\) \(M\)-matrix \(P = (p_{ij})\) by replacing \(p_{ii}\) with \(X_S^{(i)} (P)\) which makes \(P\) become a singular \(M\)-matrix. If \(X(A;S;i) + X(B;S;i)\) is an \(M\)-matrix (possibly singular) for each \(i \in S\), then \[ \text{det} (A + B)^{1/n} \leq \text{det} A^{1/n} + \text{det} B^{1/n} \] holds.
scientific article; zbMATH DE number 841761