Discrete weighted Hardy Inequality in 1-D

Abstract

In this paper we consider a weighted version of one dimensional discrete Hardy's Inequality on half-line with power weights of the form n α . Namely we consider:We prove the above inequality when α ∈ [0, 1) ∪ [5, ∞) with the sharp constant c(α). Furthermore when α ∈ [1/3, 1) we prove an improved version of (0.1) by adding infinitely many positive lower order terms on the RHS of inequality (0.1). More precisely we provefor non-negative constants b k (α).

One-dimensional discrete Hardy and Rellich inequalities on integers

“…In [21], the inequality was proved when α P p0, 1q. In a very recent paper [8], author proved (1.2) when α ą 5. The novelty of this paper is two fold.…”

“…We will prove inequality (1.2) with the sharp constant when α is an even natural number. This problem has been considered previously in the papers [21], [8]. To our surprise, this simple looking inequality hasn't been proved yet for all real numbers α.…”

“…Till date, various proofs of Hardy inequalities exist, most recent ones being [13], [6], [20], [18], [17]. It is worthwhile to mention papers [8], [11], [14], [2], [7], [4], [23], [22], [3], [14], [15], [16], where various variants of inequality (1.1) have been studied and applied.…”

Symmetrization inequalities on one-dimensional integer lattice

“…In the last section 5, we apply the weighted Polya-Szegö inequality proved in section 2 to prove discrete Hardy's inequality on non-negative integers with weights n α for 1 < α ≤ 2, refer [10] and [11] for the proof of this inequality for α ∈ [0, 1) ∪ [5, ∞) and α ∈ 2N respectively.…”