A note on a determinantal inequality
Let \(\| \cdot \|\) denote an operator norm on the set \(M_ n({\mathbb{C}})\) of \(n\times n\) matrices over the complex numbers, and let \(A,B\in M_ n({\mathbb{C}})\). Then the following inequality holds: \[ | \det (B)-\det (A)| \leq \frac{\| B\|^ n-\| A\|^ n}{\| B\| -\| A\|}\| B-A\|. \] Equality will be attained for scalar matrices A and B of the same sign. This improves an earlier result by the same author [Linear Multilinear Algebra 12, 81-98 (1982; Zbl 0491.15002)].
scientific article; zbMATH DE number 4163082