Equivalent Properties of Two Kinds of Hardy-Type Integral Inequalities

Michael Th Rassias,Andrei Raigorodskii,Bicheng Yang

doi:10.3390/sym13061006

Abstract

In this paper, using weight functions as well as employing various techniques from real analysis, we establish a few equivalent conditions of two kinds of Hardy-type integral inequalities with nonhomogeneous kernel. To prove our results, we also deduce a few equivalent conditions of two kinds of Hardy-type integral inequalities with a homogeneous kernel in the form of applications. We additionally consider operator expressions. Analytic inequalities of this nature and especially the techniques involved have far reaching applications in various areas in which symmetry plays a prominent role, including aspects of physics and engineering.

Highlights

Academic Editor: Ioan Ras, a Received: 13 May 2021Accepted: 1 June 2021In 1925, by introducing one pair of conjugate exponents ( p, q), Hardy [1] established a well-known extension of Hilbert’s integral inequality as follows.If p > 1, 1p + 1q = 1, f ( x ), g(y) ≥ 0, Published: 4 June 2021 0
This completes the proof of the Lemma
Define a Hardy-type integral operator of the second kind with the homogeneous kernel: T2

Summary

Introduction

In 1925, by introducing one pair of conjugate exponents ( p, q), Hardy [1] established a well-known extension of Hilbert’s integral inequality as follows. In 1998, by introducing an independent parameter λ > 0, Yang proved an extension of Hilbert’s integral inequality with the kernel ( x+1y)λ (cf [5,6]). Z 1 h(u)uσ−1 du = φ1 (σ ) ∈ R+ , and (4) reduces to the following Hardy-type integral inequality with nonhomogeneous kernel: Symmetry 2021, 13, 1006. If h( xy) = 0, for xy < 1, : h(u)uσ−1 du = φ2 (σ ) ∈ R+ , and (4) reduces to the following kind of Hardy-type integral inequality with nonhomogeneous kernel:. In this paper, using weight functions as well as employing various techniques from real analysis, we establish a few equivalent conditions of two kinds of Hardy-type integral inequalities with the nonhomogeneous kernel:. Hardy-type integral inequalities with a homogeneous kernel in the form of applications

Two Lemmas

Main Results and Corollaries

Operator Expressions