In 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.

中文翻译：

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一维尖锐离散 Hardy-Rellich 不等式

在本文中，我们建立离散 Hardy-Rellich 不等式和和最佳常数，对于任何。据我们所知，这些严重的不平等对于。我们的方法是使用加权等式通过移动权重获得一些尖锐的 Hardy 不等式，然后通过迭代解决高阶情况。我们还提供了一个新的 Hardy–Leray 型不等式与连续设置相同的常数。此外，主要思想也适用于一般图表或 环境。