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On improvements of the Hardy, Copson and Rellich inequalities

Bikram Das, Atanu Manna

Abstract

Using a method of factorization and by introducing a generalized discrete Dirichlet's Laplacian matrix $(-Δ_Λ)$, we establish an extended improved discrete Hardy's inequality and Rellich inequality in one dimension. We prove that the discrete Copson inequality (E.T. Copson, \emph{Notes on a series of positive terms}, J. London Math. Soc., 2 (1927), 9-12.) in one-dimension admits an improvement. We also prove that the improved Copson's weights are optimal (in fact \emph{critical}). It is shown that improvement of the Knopp inequalities (Knopp in J. London Math. Soc. 3(1928), 205-211 and 5(1930), 13-21) lies on improvement of the Rellich inequalities. Further, an improvement of the generalized Hardy's inequality (Hardy in Messanger of Math. 54(1925), 150-156) in a special case is obtained.

On improvements of the Hardy, Copson and Rellich inequalities

Abstract

Using a method of factorization and by introducing a generalized discrete Dirichlet's Laplacian matrix , we establish an extended improved discrete Hardy's inequality and Rellich inequality in one dimension. We prove that the discrete Copson inequality (E.T. Copson, \emph{Notes on a series of positive terms}, J. London Math. Soc., 2 (1927), 9-12.) in one-dimension admits an improvement. We also prove that the improved Copson's weights are optimal (in fact \emph{critical}). It is shown that improvement of the Knopp inequalities (Knopp in J. London Math. Soc. 3(1928), 205-211 and 5(1930), 13-21) lies on improvement of the Rellich inequalities. Further, an improvement of the generalized Hardy's inequality (Hardy in Messanger of Math. 54(1925), 150-156) in a special case is obtained.
Paper Structure (7 sections, 4 theorems, 101 equations)

This paper contains 7 sections, 4 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Further extension of improved Hardy's inequality
  3. Improvement of the Copson inequality
  4. Extension of improved Rellich inequality
  5. Improvement of Knopp inequality via Rellich inequality
  6. Knopp inequality of order $\alpha=2$
  7. Knopp inequality of order $\alpha\geq 1$

Key Result

Theorem 2.1

Let $A=\{A_n\}$ be any sequence of complex numbers such that $A_n\in C_c(\mathbb{N}_0)$ with $A_{0}=0$ and $\mu=\{\mu_{n}\}$ be any strictly positive sequence of real numbers. Then we have the following identity: where the sequence $\eta=\{\eta_{n}\}$ is defined as below: and $R^{(1)}_{\Lambda}$ has the following matrix representation: The identity (HFI) means the following inequality holds:

Theorems & Definitions (17)

  • Theorem 2.1
  • proof
  • proof
  • Theorem 3.1
  • proof
  • proof
  • proof
  • proof
  • Theorem 3.2
  • proof
  • ...and 7 more