One‐dimensional sharp discrete Hardy–Rellich inequalities

Abstract

AbstractIn this paper, we establish discrete Hardy–Rellich inequalities on with and optimal constants, for any . As far as we are aware, these sharp inequalities are new for . 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 with the same constant as the continuous setting. Furthermore, the main ideas work also for general graphs or the  setting.