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})).

In this paper, we first extend the well-known inequalities to the case of sector matrices. We also explore the adjointness of operator inequalities with binary operations for sector matrices. As a result of our exploration, we establish four distinct inequalities: a matrix inequality, a unitarily invariant norm inequality, a singular value inequality, and a determinant inequality. For example, we demonstrate that if $sigma_{1}$ and $ sigma_{2} $ are non-zero connections, and if $A$, $B$, and $C$ belong to $mathcal{S}_{alpha}$,  such that

	begin{equation*}

	mathcal{R}left(A  sigma_{1}  (B  sigma_{2}  C)right) leq cos^{4}(alpha)  mathcal{R}left((A  sigma_{1}  B)  sigma_{2}  (A  sigma_{1}  C)right),

	end{equation*}

	then

	begin{equation*}

	mathcal{R}left(A  sigma_{1}^* (B  sigma_{2}^*  C)right)geq cos^{4}(alpha)  mathcal{R}left((A sigma_{1}^*   B)  sigma_{2}^*  (A  sigma_{1}^*  C)right).

	end{equation*}