Assume that ${\bf A} \in {\mathbb C}^{N\times N}$ is a positive semi-definite matrix, and $\bf B \in {\mathbb C}^{N\times N}$ is a positive definite matrix, i.e., ${\bf A} \succeq {\bf 0}$ and ${\bf B} \succ {\bf 0}$. ${\bf I}_N$ stands for the $N \times N$ dimensional identity matrix. ${\bf X} \in {\mathbb C}^{N\times N}$ is a matrix variabe and satisfies ${\bf X} \succeq {\bf 0}$. I think function \begin{equation} f({\bf X}) = \log\left|{\bf A}\left({\bf G}{\bf X}{\bf G}^H+{\bf B}\right)^{-1}+{\bf I}_N\right| \end{equation} is convex with respect to matrix $\bf X$. For the simple scalar case, \begin{equation} f(x) = \log\left(\frac{a}{|g|^2x + b} + 1\right), \end{equation} where $a \geq 0$, $b > 0$. It can be readily verified that $f(x)$ is convex. I tried to prove the convexity of $f({\bf X})$ by following similar steps in which Boyd proved the concavity of $\log |{\bf X}|$ (as attached), but failed. Could anyone provide some relevant hints or references? I would be very appreciated.
By the way, we can simplify $f({\bf X})$ as follows \begin{align} f({\bf X}) & = \log\left|{\bf G}{\bf X}{\bf G}^H+{\bf A} + {\bf B}\right| - \log\left|{\bf G}{\bf X}{\bf G}^H + {\bf B}\right|\\ & = \log\left|{\bf G}^H\left({\bf A} + {\bf B}\right)^{-1}{\bf G}{\bf X}+ {\bf I}_N\right| + \log\left|{\bf A} + {\bf B}\right| - \log\left|{\bf G}^H{\bf B}^{-1}{\bf G}{\bf X} + {\bf I}_N\right| - \log\left|{\bf B}\right|. \end{align} So we only need to prove the convexity of \begin{equation} {\hat f}({\bf X}) = \log\left|{\bf G}^H\left({\bf A} + {\bf B}\right)^{-1}{\bf G}{\bf X}+ {\bf I}_N\right| - \log\left|{\bf G}^H{\bf B}^{-1}{\bf G}{\bf X} + {\bf I}_N\right|. \end{equation}