Let $f : H\_{n\times n} (\mathbb{C}) \rightarrow \mathbb{R}$ be the function on Hermitian, positive semidefinite matrices $f(A) = \frac{M\_i (A)}{\det(A)}$ where $M\_i(A)$ is the determinant of the the submatrix formed by the $A\_{kl}$ where $1\leq k, l \leq i$, i.e., it is the $i\times i$ principal minor of $A$.
Is the function $f$ convex ? 



Convexity of a (non-symmetric) function of matrices
From [Theorem 9, of this article](http://www-isl.stanford.edu/~cover/papers/paper86.pdf) it follows that $A \mapsto \log\frac{M\_i(A)}{\det(A)}$ is convex on the set of positive definite matrices. The alleged convexity in the OP is a simple consequence of this stronger log-convexity result.


