REFINEMENTS OF TWO DETERMINANTAL INEQUALITIES FOR POSITIVE SEMIDEFINITE MATRICES.

Let A,B,C ∈ C^n×n be positive semidefinite matrices and let |A|, |B|, |C| be determinants of A, B, C ∈ C^n×n respectively. In this paper, the authors prove two determinantal inequalities |A + B + C| + |C| ≥ |A + C| + |B + C| + (2ⁿ - 2) |AB|^1/2 + 3(3^n-1 - 2ⁿ + 1) |ABC|^1/3 and |A + B + C| + |A| + |B| + |C| ≥ |A + B| + |A + C| + |B + C| + 3(3^n-1 - 2ⁿ + 1)|ABC|^1/3. These two inequalities refine known ones. [ABSTRACT FROM AUTHOR]