  About last 70s, Haynsworth [6] used a result of the Schur complement to
refine a determinant inequality for positive definite matrices. Haynsworth's
result was improved by Hartfiel [5]. We extend their result to a larger class
of matrices, namely, matrices whose numerical range is contained in a sector.
Our proof relies on a number of new relations for the Schur complement of this
class of matrices.
