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)).