One-dimensional sharp discrete Hardy–Rellich inequalities

In this paper, we establish discrete Hardy–Rellich inequalities on (Formula presented.) with (Formula presented.) and optimal constants, for any (Formula presented.). As far as we are aware, these sharp inequalities are new for (Formula presented.). Our approach is to use weighted equalities to get some sharp Hardy inequalities using shifting weights, then to settle the higher order cases by iteration. We provide also a new Hardy–Leray–type inequality on (Formula presented.) with the same constant as the continuous setting. Furthermore, the main ideas work also for general graphs or the (Formula presented.) setting.