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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 $\beta$?

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    $\begingroup$ Isn't that taken care of by letting $C=A^{-1}$, $D=B^{-1}$ and applying the $\alpha,\beta\ge 0$ case to $C,D$? $\endgroup$ Sep 3, 2014 at 17:46
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    $\begingroup$ Consider the case $A=\lambda I$ and $B=I$. Then your claim is $(\frac\alpha{\alpha+\beta}\lambda^{\alpha+\beta}+\frac\beta{\alpha+\beta})^n\ge\lambda^{n\alpha}$. I suggest playing with this special case to get an idea what might happen. If $\alpha=2$ and $\beta=-1$, this estimate becomes $(2\lambda-1)^n\ge\lambda^{2n}$, which is certainly false for $\lambda$ large enough. $\endgroup$ Sep 3, 2014 at 17:59

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