Assume that $n$ hermitian positive-definite matrices $H_1, H_2, ..., H_n$ are given. Also, we consider arbitrary subset of these matrices $H_{i_1}, ..., H_{i_k}$, $i_1\neq...\neq i_k \in \{1,..,n\}$, $k \leq n$.
The question is, if the next statement always holds and how can it be proved:
$$ tr \left[(H_1 + ... + H_n)^{-1}\right] \leq tr\left[(H_{i_1} + ... + H_{i_k})^{-1}\right] $$
So far I have checked it using 2 matrices $A$ and $B$ and the property of hermitian matrices that $det(A+B) \geq det(A) + det(B)$, but it gives the condition that if $\frac{tr(B)}{det(B)} \leq \frac{tr(A)}{det(A)}$, then $tr \left[(A+B)^{-1}\right] \leq tr\left[(A)^{-1}\right]$.