adjointations of operator inequalities for sector matrices

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 σ1 and σ2 are non-zero connections, and if a, b, and c belong to sα, such that r (aσ1(bσ2c)) ≤ cos⁴ (α) r ((aσ1b)σ2(aσ1c)), then r (aσ∗1 (bσ∗2c)) ≥ cos⁴ (α) r ((aσ∗1b)σ∗2 (aσ∗1c)).