On a determinantal inequality arising from diffusion tensor imaging

In comparing geodesics induced by different metrics, Audenaert formulated the following determinantal inequality $$\det (A^2+\mathrm{\mid }BA\mathrm{\mid })\le \det (A^2+AB),$$ where $A,B$ are $n\times n$ positive semidefinite matrices. We complement his result by proving $$\det (A^2+\mathrm{\mid }AB\mathrm{\mid })\ge \det (A^2+AB)\mathrm{.}$$ Our proofs feature the fruitful interplay between determinantal inequalities and majorization relations. Some related questions are mentioned.