Extension of Matic's results (Q498325)

Recently, \textit{I. Matic} obtained two interesting inequalities involving the ratio \(\frac{\det (A+D)}{\det (A)} \), where \(A\) is a positive definite matrix and \(D\) is a positive definite block-diagonal matrix [Linear Algebra Appl. 449, 166--174 (2014; Zbl 1302.15024)]. In the present paper, the author extends these inequalities to the larger class of matrices whose numerical radii are contained in a sector.