What conditions on ${\bf{A}}_1,{\bf{A}}_2$ should be satisfied to make the formula concave over ${X_i}$

Question

$\log \frac{{\left| {{{\bf{A}}_1} + \sum\limits_{i = 1}^K {{\bf{H}}_i^H{{\bf{X}}_i}{{\bf{H}}_i}} } \right|}}{{\left| {{{\bf{A}}_2} + \sum\limits_{i = 1}^K {{\bf{H}}_i^H{{\bf{X}}_i}{{\bf{H}}_i}} } \right|}}$,where ${{\bf{X}}_i} \in {C^{m \times m}},{{\bf{H}}_i} \in {C^{m \times n}},{{\bf{A}}_1},{{\bf{A}}_2} \in {C^{n \times n}}$.And $\left| \cdot \right|$ represents the determinant of a square matrix. I'm looking forward to your kind help!!