In the theory of matrix operators the sharp estimates of their norms in various spaces of sequences are of great importance. At present, finding the exact values of the norms of a discrete operator, even in the classical spaces of Lebesgue sequences, is an open problem and it is not always possible to find necessary and sufficient conditions for fulfillment various weighted estimates for some classes of matrix operators. Therefore, the establishment of weighted estimates for certain classes of matrix operators, two-sided estimates of norms are actual problems of the modern theory of discrete operators. In this paper we consider the problem of finding necessary and sufficient conditions for the fulfillment of a discrete Hilbert-Stieltjes type inequality when . Moreover, an alternative proof of a discrete Hardy-type inequality with variable limits of summation is presented