More results on singular value inequalities of matrices

The 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 give a new equivalent form and some relevant generalizations of this inequality. In particular, we show thatsjA14B34+A34B14⩽sj(A+B),j=1,…,nfor any n × n positive semidefinite matrices A, B, which proves a special case of Zhan’s conjecture posed in 2000.