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Sharp Hardy's Inequalities in Hilbert Spaces

Dimitar K. Dimitrov, Ivan Gadjev, Mourad E. H. Ismail

Abstract

We study the behavior of the smallest possible constants $d(a,b)$ and $d_n$ in Hardy's inequalities $$ \int_a^b\left(\frac{1}{x}\int_a^xf(t)dt\right)^2\,dx\leq d(a,b)\,\int_a^b [f(x)]^2 dx $$ and $$ \sum_{k=1}^{n}\Big(\frac{1}{k}\sum_{j=1}^{k}a_j\Big)^2\leq d_n\,\sum_{k=1}^{n}a_k^2. $$ The exact constant $d(a,b)$ and the precise rate of convergence of $d_n$ are established and the extremal function and the ``almost extremal'' sequence are found.

Sharp Hardy's Inequalities in Hilbert Spaces

Abstract

We study the behavior of the smallest possible constants and in Hardy's inequalities and The exact constant and the precise rate of convergence of are established and the extremal function and the ``almost extremal'' sequence are found.
Paper Structure (4 sections, 5 theorems, 75 equations)

This paper contains 4 sections, 5 theorems, 75 equations.

Table of Contents

  1. Introduction and statement of the results
  2. Proof of Theorem \ref{['th1']}
  3. Proof of Theorem \ref{['th2']}
  4. Hardy's inequality \ref{['eq14']}, the zeros of continuous dual Hahn polynomials and the proof of Theorem \ref{['Th1.4']}

Key Result

Theorem 1.1

Let $a$ and $b$ be any fixed numbers with $0<a<b<\infty$. Then the inequality where $\alpha$ is the only solution of the equation holds for every $f \in L^2[a,b]$. Moreover, equality in MR1 is attained for

Theorems & Definitions (5)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem A