Refinement multidimensional dynamic inequalities with general kernels and measures

Samir H. Saker,Haytham M. Rezk,Islam Abohela,Dumitru Baleanu

doi:10.1186/s13660-019-2255-8

Abstract

Using the properties of superquadratic and subquadratic functions, we establish some new refinement multidimensional dynamic inequalities of Hardy’s type on time scales. Our results contain some of the recent results related to classical multidimensional Hardy’s and Pólya–Knopp’s inequalities on time scales. To show motivation of the paper, we apply our results to obtain some particular multidimensional cases and provide refinements of some Hardy-type inequalities known in the literature.

Highlights

Hardy [13] proved the integral inequality∞1 t p g(s) ds dt ≤ p p∞ gp(t) dt (1)0 t0 p–1 0 for a nonnegative integrable function g in the space Lp(R+), 1 < p < ∞, where the constant (p/(p – 1))p is the best possible
We prove some new refined multidimensional dynamic inequalities of Hardy type with weighted functions and nonnegative kernel using the properties of superquadratic functions
To show motivation of the paper, we will apply our results to obtain some particular multidimensional cases and provide refinements of some Hardy-type inequalities known in the literature

Summary

Introduction

0 t0 p–1 0 for a nonnegative integrable function g in the space Lp(R+), 1 < p < ∞, where the constant (p/(p – 1))p is the best possible. They proved that if 0 < b ≤ ∞, λ : (0, b) → R is a nonnegative function such that the function t → λ(t)/t2 is locally integrable in (0, b), and Ψ is a convex function on (a, c), where –∞ ≤ a ≤ c ≤ ∞, b λ(t)Ψ b dt η(t)Ψ g(t). In [18] the authors obtained some new Hardy-type inequalities on time scales by using the notion of superquadratic functions They proved that if (Ω1, Σ1, μ1) and (Ω2, Σ2, μ2) are two time scale measure spaces with positive σ -finite measures, λ : Ω1 →.

Ak g

Proof We start with the identity