On the Schur product of H-matrices and non-negative matrices, and related inequalities

1. Introduction. Let ℛn denote the set of all n × n matrices with real elements, and let denote the subset of ℛn consisting of all real, n × n, symmetric positive-definite matrices. We shall use the notation to denote that minor of the matrix A = (aij) ∈ ℛn which is the determinant of the matrix The Schur Product (Schur (14)) of two matrices A, B ∈ ℛn is denned by where A = (aij), B = (bij), C = (cij) and Let ϕ be the mapping of ℛn into the real line defined by for all A ∈ ℛn, where, as in the sequel, .