Some Determinant Inequalities for Two Positive Definite Matrices Via a result of Cartwright and Field

Abstract

In this paper we prove among others that, if the positive definite matrices \(A, B\) satisfy the condition \(A \leq B\), then \[ (0 \leq) \frac{1}{12} \left[ [\det(A)]^{-1} - 2 [\det(B)]^{-1} + [\det(2B - A)]^{-1} \right] \] \[ \leq \frac{[\det(B)]^{-1} + [\det(A)]^{-1}}{2} - \int_{0}^{1} [\det((1 - t)B + tA)]^{-1} \, dt. \] If \(A \leq B < 2A\), then also \[ \frac{[\det(B)]^{-1} + [\det(A)]^{-1}}{2} - \int_{0}^{1} [\det((1 - t)B + tA)]^{-1} \, dt \] \[ \leq \frac{1}{12} \left[ [\det(2A - B)]^{-1} - 2 [\det(A)]^{-1} + [\det(B)]^{-1} \right]. \]