On Bicheng‐Debnath′s generalizations of Hardy′s integral inequality

Aleksandra Cižmešija,Josip Pecaric

doi:10.1155/s0161171201005816

Abstract

We consider Hardy′s integral inequality and we obtain some new generalizations of Bicheng‐Debnath′s recent results. We derive two distinguished classes of inequalities covering all admissible choices of parameter k from Hardy′s original relation. Moreover, we prove the constant factors involved in the right‐hand sides of some particular inequalities from both classes to be the best possible, that is, none of them can be replaced with a smaller constant.

Highlights

In the 1920s, Hardy proved the following integral inequality: let p, k ∈ R, p > 1, k ≠ 1, and for x ∈ (0, ∞) denote F (x) = x f (t)dt, ∞f (t)dt, x k > 1, k < 1, (1.1)where f is a nonnegative measurable function such that x1−k/pf ∈ Lp(0, ∞)
We will make a detailed analysis of the weight functions appearing on the right-hand sides of all derived inequalities, what will be used in proving that the obtained constant factor (p/|k − 1|)p is the best possible in cases where a = 0 or b = ∞
That fact will be very helpful in analyzing strict inequalities for some particular parameters a and b from the statement of the lemma in what follows: suppose that a = 0 or b = ∞, that is, consider the intervals [0, b) and [a, ∞)

Summary

Introduction

∞, the inequality bb p b f (t)dt dx < pp hp,a,b(x)xpf p(x)dx ax a holds, where the weight function hp,a,b is hp,a,b(x) = Note that Bicheng-Debnath results consider only two particular cases of the parameter k from inequality (1.2): k = p in (1.3), and k = 0 in (1.5).

Objectives

Results

Conclusion