On Drury's solution of Bhatia & Kittaneh's question

Let A,B be n×n positive semidefinite matrices. Bhatia and Kittaneh asked whether it is trueσj(AB)≤12λj(A+B),j=1,…,n where σj(⋅), λj(⋅), is the j-th largest singular value, eigenvalue, respectively. The question was recently solved by Drury in the affirmative. This article revisits Drury's solution. In particular, we simplify the proof for a key auxiliary result in his solution.