Let $A\in \mathbb{R}^{n\times n}$ be a positive definite symmetric matrix with eigenvalues $\lambda_1\ge \cdots\ge \lambda_n$, $X\in \mathbb{R}^{n\times k}$ such that $X'X=I_k$ ($X'$ means the transpose of $X$) and $n\ge 2k$, then $$\det(X'AXX'A^{-1}X)\le \prod\limits_{j=1}^k\frac{(\lambda_j+\lambda_{n-j+1})^2}{4\lambda_j\lambda_{n-j+1}}.$$

This result was first proved by Bloomfield and Watson(1975) and Knott(1975). I came across a note by H. Yang (A brief proof on the generalized variance bound of the relative efficiency in statistics, Communications in Statistics - Theory and Methods, 19(1990):12, 4587-4590), but the notation of his proof was rather confusing. Can any one make a clean proof or explain his proof?

(I don't have an e-version of that paper, sorry for inconvenience)