<p>This work studies tensor inequalities within the framework of the tensor C-product in three main parts. First, several classical matrix inequalities, such as the Löwner–Heinz and Young inequalities, are generalized to tensors. Second, the tensor norm inequalities under the C-product are investigated. Many fundamental results on tensor Frobenius and spectral norms within the framework of the tensor C-product are presented, and then the arithmetic-geometric mean inequality and its general cases, the Hölder and Minkowski inequalities, for tensors, are presented. Last, we generalize the key matrix eigenvalue results, including the Gershgorin circle theorem, Schur inequality, Bauer–Fike theorem, and Hoffman–Wielandt theorem on C-eigenvalues, to the C-product domain.</p>