For positive semidefinite matrices $A,B,C \in \mathbb{R}^{n\times n}$, the following inequalities are well known:

$$(\det(A+B))^{1/n} \geq (\det A)^{1/n} + (\det B)^{1/n} $$

and

$$\det(A+B+C) + \det(C) \geq \det(A+C) + \det(B+C).$$

The first is, of course, Minkowski's determinant inequality. I'm not sure whether the second has a name, but it can be found, e.g., here and here.

I would like to know whether anyone knows of any reverse forms of these inequalities. To be more specific, I note that the Minkowski determinant inequality can be viewed as a consequence of the Entropy Power Inequality applied to Gaussian random vectors with covariance matrices $A$ and $B$. Applying similar reasoning to the entropy power inequality appearing in Theorem 4 of this paper, we may conclude that the following reverse counterparts to the above inequalities hold:

$$(\det(A+B))^{1/n} \leq (\det A)^{1/n}\left(\frac{\frac{1}{n}\operatorname{Tr}(B^{-1})}{(\det B^{-1})^{1/n}} \right) + (\det B)^{1/n}\left(\frac{\frac{1}{n}\operatorname{Tr}(A^{-1})}{(\det A^{-1})^{1/n}} \right),~~~~~(\star) $$

and

$$(\det(A+B+C) \det(C) )^{1/n} + (\det(A) \det(B) )^{1/n} \leq (\det(A+C) \det(B+C) )^{1/n} .~~~(\star\star)$$

The first may be obtained from the second by taking $C = \varepsilon I$ and letting $\varepsilon \downarrow 0$. I view $(\star)$ as a reverse counterpart to Minkowski's determinant inequality, since the ratio

$$\frac{\frac{1}{n}\operatorname{Tr}(B^{-1})}{(\det B^{-1})^{1/n}}\geq 1$$

by the AM-GM inequality applied to the eigenvalues of $B^{-1}$ (and similarly for the ratio involving $A^{-1}$), with near equality if the matrix $B$ has eigenvalues all roughly the same.

Has anyone seen these inequalities before, or does anyone have a direct proof of $(\star\star)$?