Extensions of Fischer's inequality

Denote by B(n(1), ..., n(k)) the set of block matrices whose (i, j)-blocks are n(i) x n(j) complex matrices. Let A(i) is an element of B(n(1), ..., n(k)) be positive semidefinite and D-i is an element of B(n(1), ..., n(k)) be block diagonal matrices for 1 <= i <= m. We obtain the following extension of Fischer's inequality: det (Sigma(m)(i=1) D(i)A(i)(pi) D-i(*)) <= Pi(k)(j=1) det (Sigma(m)(i=1)[D-i](j)[A(i)](j)(pi)[D-i](j)(*), 0 <= p(i) <= 1, where [A(i)](j) is the j-th main diagonal block of A(i). In addition, if A(i) and D-i, are nonsingular, the reverse inequality holds when -1 <= p(i) <= 0. We also extend these two results to a larger class of matrices, namely, matrices whose numerical ranges are contained in a sector. (C) 2019 Elsevier Inc. All rights reserved.