Here is the statement :
Let $A\in \mathcal{S}_n^{++}(\mathbb{R})$ and $B \in \mathcal{S}_n^{+}(\mathbb{R})$ then we have the following inequality : $(\det(A+B))^{\frac{1}{n}}\ge (\det(A))^{\frac{1}{n}}+(\det(B))^{\frac{1}{n}}$, $\ n\ge 1$.
Is it a well-known inequality and are there references about it ?
NB : $\mathcal{S}_n^{++}(\mathbb{R})$ is the set of symmetric matrices whose eigenvalues are $>0$ and $\mathcal{S}_n^{+}(\mathbb{R})$ is the set of the matrices who eigenvalues are $\ge 0$.
Thanks in advance !