On some refinement of the Cauchy–Schwarz inequality

If A and B are positive semidefinite operators on a Hilbert space and if σ is an operator mean in the sense of Kubo and Ando, then the operator inequality(A#B)⊗(A#B)⩽12{(AσB)⊗(Aσ⊥B)+(Aσ⊥B)⊗(AσB)}⩽12{(A⊗B)+(B⊗A)}holds. This inequality is a generalization of some refinement of the Cauchy-Schwarz inequality.