Some new Hardy-type inequalities on time scales

Ahmed A El-Deeb,Hamza A Elsennary,Dumitru Baleanu

doi:10.1186/s13662-020-02883-8

Ahmed A El-Deeb, Hamza A Elsennary + Show 1 more

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https://doi.org/10.1186/s13662-020-02883-8

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Abstract

In this paper, we will prove some new dynamic inequalities of Hardy-type on time scales. Some of the integral and difference inequalities that will be derived from our results in the continuous and discrete cases are original. The main results will be proved by using the dynamic Hölder inequality, integration by parts formula on time scales, and Keller’s chain rule on time scales. We will apply the main results to the continuous calculus, discrete calculus, and q-calculus as special cases.

Highlights

IntroductionIn 1920, Hardy [13] established the following renowned discrete inequality. ap(n)
In 1920, Hardy [13] established the following renowned discrete inequality.Theorem 1.1 If {a(n)}∞ n=0 is a nonnegative real sequence and p > 1, ∞ n a(m) p ≤ p p∞ ap(n).n p–1 n=1 m=1 n=1 (1.1)Hardy discovered this inequality while attempting to sketch an easier proof of Hilbert’s inequality for double series which was known at that time.In 1925, using the calculus of variations, Hardy himself in [7] gave the integral analogue of inequality (1.1) as follows: Theorem 1.2 If f is a nonnegative continuous function on [0, ∞) and p > 1
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 may be an arbitrary closed subset of the real numbers R, see [8, 10]

Summary

Introduction

In 1920, Hardy [13] established the following renowned discrete inequality. ap(n). In 1925, using the calculus of variations, Hardy himself in [7] gave the integral analogue of inequality (1.1) as follows: Theorem 1.2 If f is a nonnegative continuous function on [0, ∞) and p > 1, . In 1928, Hardy [15] established the continuous versions of inequalities (1.3) and (1.4) as follows: Theorem 1.5 Let f be a nonnegative continuous function on [0, ∞). 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 may be an arbitrary closed subset of the real numbers R, see [8, 10]. We will establish some original dynamic inequalities of Hardy-type on time scales which may be considered as generalizations of inequalities (1.15), (1.16), (1.17), and (1.18).

Main results

Conclusions