Convexity proof of log-determinant $\log\left|{\bf A}\left({\bf G}{\bf X}{\bf G}^H+{\bf B}\right)^{-1}+{\bf I}_N\right|$ with respect to $\bf X$ 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}

 A: It has been proven in [1] that for given ${\bf A} \succeq {\bf 0},$ $h ({\bf X}) = \log \det({\bf A} {\bf X}^{-1} + {\bf I}_N)$ is convex over the positive semidefinite cone.
We then show that $h ({\bf X})$ is a monotone nonincreasing function, i.e., for any ${\bf X}_1 \succeq {\bf X}_2 \succeq {\bf 0}$, $h ({\bf X}_1) \leq h ({\bf X}_2)$.
If ${\bf X}_1 \succeq {\bf X}_2 \succeq {\bf 0}$, we know that ${\bf A} {\bf X}_1^{-1} + {\bf I}_N \preceq {\bf A} {\bf X}_2^{-1} + {\bf I}_N$.
Due to the monotonicity of $\log \det(\cdot)$, we have $h ({\bf X}_1) \leq h ({\bf X}_2)$.
Hence, $h ({\bf X})$ is a monotone nonincreasing function.
Let $g({\bf X}) = {\bf G} {\bf X} {\bf G}^H + {\bf B} $, which is an affine function, and $f({\bf X}) = h(g({\bf X}))$.
Then, from the condition which guarantees convexity of a composite function, [2, Subsection 3.2.4], it is known that $f({\bf X})$ is convex over the positive semidefinite cone.
[1]. Kim, KK.K., ''Optimization and Convexity of $\log \det({\bf A} {\bf X}^{-1} + {\bf I}_N)$'', Int. J. Control Autom. Syst., pp. 1067–1070, 2019.
[2]. S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge university press, 2004.
