Somayeh Malekinejad - Payame Noor University

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^{-۱}left( A sigma B)right)leq sec^۲alpha~mathcal{R} (Phi(A^{-۱})sharp Phi(B^{-۱})).

Callebaut inequality, Positive linear map, Sector matrices, Semi-self-adjoint mean.