Assume that A and B are symmetric, positive definite matrices of the same size.
For which set of real parameters $\alpha $ and $\beta$ the following relation holds $\det(\frac{\alpha}{\alpha+\beta}A^{\alpha+\beta}+\frac{\beta}{\alpha+\beta}B^{\alpha+\beta})≥det(A)^{\alpha}det(B)^{\beta}$
It is relatively easy to prove that this true if both $\alpha$ and $\beta$ are nonnegative. however how to prove this relation for negative values of $\alpha$ and $\beta$?