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Summary
In this paper, we establish discrete Hardy-Rellich inequalities on N$\mathbb {N}$ with Delta l2$\Delta <^>\frac{\ell }{2}$ and optimal constants, for any l > 1$\ell \geqslant 1$. As far as we are aware, these sharp inequalities are new for l > 3$\ell \geqslant 3$. 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 N${\mathbb {N}}$ with the same constant as the continuous setting. Furthermore, the main ideas work also for general graphs or the lp$\ell <^>p$ setting.
