The arithmetic-geometric mean inequality for singular values due to Bhatia and Kittaneh says that 2sj(AB*) ≤ sj(A*A + B*B), j = 1, 2, ... for any matrices A, B. We first give new proofs of this inequality and its equivalent form. Then we use it to prove the following trace inequality: let A0 be a positive definite matrix and A 1,..., Ak be positive semidefinite matrices. Then tr ∑j=1k (∑i=0jA i-2) Aj < tr A0-1. © 2003 Elsevier Inc. All rights reserved.
Zhan, X. (2004). On some matrix inequalities. Linear Algebra and Its Applications, 376(1–3), 299–303. https://doi.org/10.1016/j.laa.2003.08.008