Determinantal and eigenvalue inequalities for matrices with numerical ranges in a sector

Let A=(A11A12A21A22)∈Mn, where A11∈Mm with m⩽n/2, be such that the numerical range of A lies in the set {eiφz∈C:|ℑz|⩽(ℜz)tanα}, for some φ∈[0,2π) and α∈[0,π/2). We obtain the optimal containment region for the generalized eigenvalue λ satisfyingλ(A1100A22)x=(0A12A210)xfor some nonzero x∈Cn, and the optimal eigenvalue containment region of the matrix Im−A11−1A12A22−1A21 in case A11 and A22 are invertible. From this result, one can show |det(A)|⩽sec2m(α)×|det(A11)det(A22)|. In particular, if A is an accretive-dissipative matrix, then |det(A)|⩽2m|det(A11)det(A22)|. These affirm some conjectures of Drury and Lin.