I need to prove that the following function is convex:
$$f(X) = -\log\left( \sqrt{\det(A - X^2)} - c \right)$$
where $X^2, A$ are positive definite $2\times 2$ matrices and $A > X^2$ in the Semidefinite sense and $c$ is a scalar such that the term under logarithm is always positive.
I have looked into similar problems in Boyd's book like the log-determinant example of section 3.1.5 which proves $\log \det(X)$ is concave, but the proof there does not directly apply here because of the $c$ term I cannot turn this into summation over eigenvalues.
One other approach is to use the exercise 3.18 of Boyd which says $\det(X)^{1/n}$ is concave for $X\in S_{++}^n$. Therefore, saying because $A - X^2$ is concave and decreasing then the term inside the $\log$ is convex. Finally because $-\log$ is convex then the whole function is convex. Is this argument correct? Also I am not entirely sure how to prove 3.18 so is it possible to provide some explanation for the whole problem? I would appreciate any help or guidance. Thanks.