Two determinantal inequalities for positive semidefinite matrices

Let A,B,C ∈ Cnxn be positive semidefinite matrices. In this paper, the authors prove two determinantal inequalities|A+B+C|+|C|≥|A+C|+|B+C|+(3n −2 n+1+1)|ABC|1/3and|A+B+C|+|A|+|B|+|C|≥|A+B|+|A+C|+|B+C|+3(3n−1−2n+1)|ABC|1/3.These two inequalities improve known ones.