An Improved Discrete p-Hardy Inequality

One‐dimensional sharp discrete Hardy–Rellich inequalities

“… $$$$\begin{align} \sum _{x\in X} V(x) |\nabla u|^p_p(x)\geqslant - \sum _{x\in X}\frac{{\rm div}{\left[V (\nabla f)_b^{p-1}\right]}}{f^{p-1}}(x) u^{p}(x), \quad \forall \; u \in C_c(X). \end{align}$$$$The proof of [6, Proposition 3] used somehow an equivalent form of () with $$V \equiv 1$$ over $${\mathbb {N}}$$, see also [5, Theorem 3.1] for locally summable graphs.…”

“…Inspired by [4], they showed that the weights $$(w_n)$$ are critical in many aspects, in particular the inequality () cannot hold with any sequence $$\gneq w_n$$. See also [6] for similar improvement of () with general $$p > 1$$.…”