NEW INEQUALITIES INVOLVING OPERATOR MEANS FOR SECTOR MATRICES

The main goal of this paper is to discuss the Callebaut inequality and mean-convex inequality from positive definite matrices to sector matrices in a more general setting. Afterward, several inequalities involved positive linear map, are presented for sector matrices. For instance, we show that A, B ∈ Sα are two sector matrices, then for all σ ≥ ♯ we have R(Φ−1 (AσB)) ≤ sec2 α R(Φ(A−1)♯Φ(B−1)).