The statistical convergence of double sequences was presented by Mursaleen-Edely and Tripathy in two ways, [10, 16]. The statistical core of double sequences was introduced by Cakan- Altay, . The sigma-convergence and sigma-core of double sequences were defined by Cakan-Altay-Mursaleen, . In this paper, we will study some new inequalities related to the Pringsheim, statistical and sigma-cores of double sequences. To achieve this goal, we will characterize some classes of four-dimensional matrices.