  Yuan and Leng (2007) gave a generalization of Ky Fan's determinantal
inequality, which is a celebrated refinement of the fundamental Brunn-Minkowski
inequality $(\det (A+B))^{1/n} \ge (\det A)^{1/n} +(\det B)^{1/n}$, where $A$
and $B$ are positive semidefinite matrices. In this note, we first give an
extension of Yuan-Leng's result to multiple positive definite matrices, and
then we further extend the result to a larger class of matrices whose numerical
ranges are contained in a sector. Our result improves a recent result of Liu
[Linear Algebra Appl. 508 (2016) 206--213].
