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*}
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