Weak log majorization and determinantal inequalities

Denote by ${\mathrm{\P }}_n$ the set of $n\times n$ positive definite matrices. Let $D = D_1\oplus \dots \oplus D_k$,where$D_1\in \P_{n_1}, \dots, D_k \in \P_{n_k}$with$n_1+\cdots + n_k=n$\mathrm{.}Partition$C\in \P_n$accordingto$(n_1, \dots, n_k)$sothat$\Diag C = C_1\oplus \dots \oplus C_k$. We prove the following weak log majorization result: \begin{equation*} \lambda (C^{-1}_1D_1\oplus \cdots \oplus C^{-1}kD_k)\prec{w ,\log} \lambda(C^{-1}D), \end{equation*} where $\lambda (A)$ denotes the vector of eigenvalues of $A\in \text{Cnn}$. The inequality does not hold if one replaces the vectors of eigenvalues by the vectors of singular values, i.e., \begin{equation*} s(C^{-1}_1D_1\oplus \cdots \oplus C^{-1}kD_k)\prec{w ,\log} s(C^{-1}D) \end{equation*} is not true. As an application, we provide a generalization of a determinantal inequality of Matic \cite[Theorem 1.1]{M}. In addition, we obtain a weak majorization result which is complementary to a determinantal inequality of Choi \cite[Theorem 2]{C} and give a weak log majorization open question.