On some matrix inequalities

AbstractThe arithmetic–geometric mean inequality for singular values due to Bhatia and Kittaneh says that2sj(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. Thentr∑j=1k∑i=0jAi−2Aj