Integral Inequality of Hardy-Type on Time Scales

Emmanuella Ehui Aribike,Kamilu Rauf,Yisa Oluwatoyin Anthonio,Michael Oyelami Ajisope

doi:10.4236/apm.2019.96025

Abstract

Inequalities are essential in the study of Mathematics and are useful tools in the theory of analysis. They have been playing a critical role in the study of the existence and uniqueness properties of solutions of initial and boundary value problems for differential equations as well as difference equations with their bounds. In this paper, we obtain new integral inequalities mainly by using some known inequalities. Various generalizations of Hardy's inequality are special cases of the results therein.

Highlights

Inequalities are essential in the study of Mathematics and are useful tools in the theory of analysis
They have been playing a critical role in the study of the existence and uniqueness properties of solutions of initial and boundary value problems for differential equations as well as difference equations with their bounds
Some inequality were in their reigns such as Wirtinger’s, Holder’s, Cauchy’s, Minkwoski’s, Hardy’s and Opial’s inequalities

Summary

Introduction

Some inequality were in their reigns such as Wirtinger’s, Holder’s, Cauchy’s, Minkwoski’s, Hardy’s and Opial’s inequalities. Let η be continuous and non-decreasing on [α, β ] , 0 ≤α ≤ β < ∞ , with η(x) > 0 for x > 0 , 1≤σ ≤ q < ∞ and f (x) be non-negative Lebesgue-Stieltjes integrable with respect to η ( x) on [α, β ]. Oguntuase [6] presented an integral inequality by using Hölder’s inequality to obtain an integral inequality that has Opial’s and Hardy’s inequalities as special cases He observed that constant (1.12) at the right hand side was wrongly written and obtained a better constant as stated below: Theorem 1.13. We shall show that Theorem 1.1 in its modified form leads to some extensions, variants and a new generalization of a class of inequalities which are related to Hardy’s and Opial’s integral inequalities. The validity of the left hand side of (1.19) solely depends on the right hand side

Statement of the Main Result

Remark

Discussion of the Results