Prove or disprove. Let $X$ and $Y$ be positive semidefinite matrices of size $M$ and $K$, respectively. Further , let $A$ be any $M\times K$ complex matrix. Then, the function $$-\log\det(I + XAYA^H)$$ is (jointly) convex in $X$ and $Y$.
Note. If it helps, $X$ and $Y$ can be further restricted to being diagonal.
Hint: $$\log\det M= \text{tr}(\log M)$$ Here, $M=I+XAYA^H$ is surely positive-definite. Also note that: $$\log(I+X)=\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}X^n$$