On some matrix inequalities

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 A1,…,Ak be positive semidefinite matrices. Then