Some reverse inequalities of Hardy type on time scales

A A El-Deeb,Zareen A Khan,H A El-Sennary

doi:10.1186/s13662-020-02857-w

A A El-Deeb, Zareen A Khan + Show 1 more

Open Access

https://doi.org/10.1186/s13662-020-02857-w

Copy DOI

Abstract

In this article, we obtain some new dynamic inequalities of Hardy type on time scales. The main results are derived using Fubini’s theorem and the chain rule on time scales. We apply the main results to the continuous calculus, discrete calculus, and q-calculus as special cases.

Highlights

In 1920, Hardy [15] proved the following result.Theorem 1.1 Let {a(n)}∞ n=1 be a sequence of nonnegative real numbers
In 1928, Hardy [17] proved a generalization of integral inequality (1.2) in the following theorem
The general idea is to prove a result for a dynamic equation or a dynamic inequality where the domain of the unknown function is a so-called time scale T, which is defined as an arbitrary closed subset of the real numbers R, see [9, 10]

Summary

Introduction

In 1920, Hardy [15] proved the following result.Theorem 1.1 Let {a(n)}∞ n=1 be a sequence of nonnegative real numbers. In 1928, Hardy [17] proved a generalization of integral inequality (1.2) in the following theorem. In 2002, Kaijser et al [22], using the convex functions, established a generalization of the integral Hardy inequality (1.2) in the following form.

Results

Conclusion