Improvement of some discrete Hardy inequalities with variants

Abstract In this paper, we establish a new version of one-dimensional discrete improved Hardy’s inequality with shifts by introducing a shifting discrete Dirichlet’s Laplacian. We prove that the general discrete Hardy’s inequality as well as its variants in some special cases admit improvements. Further, it is proved that two-variable discrete $p$ -Hardy inequality can also be improved via improved discrete $p$ -Hardy inequality in one dimension. The result is also extended to the multivariable cases.