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 if $ A,Bin {{mathcal S}_{alpha}}$ are two sector matrices, then for all $sigmageqsharp$ we have

begin{equation*}

mathcal{R}(Phi^{-1}left( A sigma B)right)leq sec^2alpha~mathcal{R} (Phi(A^{-1})sharp Phi(B^{-1})).