One-dimensional discrete Hardy and Rellich inequalities on integers

Abstract

In this paper, we consider a weighted version of one-dimensional discrete Hardy inequalities with power weights of the form $n^{\alpha }$. We prove the inequality when $\alpha$ is an even natural number with the sharp constant and remainder terms. We also find explicit constants in standard and weighted Rellich inequalities and its higher order versions. As a by-product of this work we derive a combinatorial identity using purely analytic methods. This suggests a correlation between combinatorial identities and functional identities.

Constant in the Lemma 3.5 corrected. This changed the constants in some of the main theorems; in particular the constant in Corollary 2.4 is 5/16 instead of previous incorrect value of 8/16. Few additional details added and some minor layout changes. This version appears in Journal of Fourier Analysis and Applications

References in corpus (2)

• A sharp form of the discrete Hardy inequality and the Keller-Pinchover-Pogorzelski inequality
• Discrete weighted Hardy Inequality in 1-D

Cited by in corpus (1)

• An optimal fractional Hardy inequality on the discrete half-line