Some reverse inequalities of Hardy type on time scales

The primary goal of this research is to prove some new Hardy-type ∇-conformable dynamic inequalities by employing product rule, integration by parts, chain rule and (γ,a)-nabla Hölder inequality on time scales. The inequalities proved here extend and generalize existing results in the literature. Further, in the case when γ=1, we obtain some well-known time scale inequalities due to Hardy inequalities. Many special cases of the proposed results are obtained and analyzed such as new conformable fractional h-sum inequalities, new conformable fractional q-sum inequalities and new classical conformable fractional integral inequalities.

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“…El-Deeb et al [16] generalized (8) that unify (4) and (5). If p ≥ 1 and γ > 1, we get Furthermore, in 2020, El-Deeb et al [17]…”

Section: Introductionmentioning

confidence: 86%

“…In this work, we prove and extend some new Hardy's inequality obtained in [15][16][17] to a standard time scale and set up numerous new sharpened types of fractional conformable ∇-integral of order γ ∈ (0, 1] on time scales. As a new work, we generalize the inequalities presented in those papers.…”

Section: Introductionmentioning

confidence: 96%

A Variety of Nabla Hardy’s Type Inequality on Time Scales

et al. 2022

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“…If k denotes the derivative of k with respect to the first variable, then f (t) = Other dynamic inequalities on time scales may be found in [37][38][39][40]. In this manuscript, we will discuss the retarded time scale case of the inequalities obtained in [1] using new techniques by replacing the upper limitς andˆ of the integral by the delay functionα(ς) ≤ ς andβ(ˆ ) ≤ˆ .…”

Section: Theorem 13 ([3])mentioning

confidence: 99%

On some new double dynamic inequalities associated with Leibniz integral rule on time scales

El‐Deeb

Rashid

2021

Adv Differ Equ

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In 2020, El-Deeb et al. proved several dynamic inequalities. It is our aim in this paper to give the retarded time scales case of these inequalities. We also give a new proof and formula of Leibniz integral rule on time scales. Beside that, we also apply our inequalities to discrete and continuous calculus to obtain some new inequalities as special cases. Furthermore, we study boundedness of some delay initial value problems by applying our results as application.

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“…The growing interest to Hardy-Copson type inequalities have taken place in the time scale calculus as well and delta unifications of these inequalities have been established in the book [4] and in the articles [59]- [65], [2,17,18,22,23] whereas their nabla counterparts and extensions can be seen in [33][34][35].…”

Section: Introductionmentioning

confidence: 99%

Diamond alpha Hardy-Copson type dynamic inequalities

Kayar

Kaymakçalan

2022

Hacettepe Journal of Mathematics and Statistics

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In this paper two kinds of dynamic Hardy-Copson type inequalities are derived via diamond alpha integrals. The first kind consists of twelve new integral inequalities which can be considered as mixed type in the sense that these inequalities contain delta, nabla and diamond alpha integrals together. The second kind involves another twelve new inequalities, which are composed of only diamond alpha integrals, unifying delta and nabla Hardy-Copson type inequalities. Our approach is quite new due to the fact that it uses time scale calculus rather than algebra. Therefore both kinds of our results unify some of the known delta and nabla Hardy-Copson type inequalities into one diamond alpha Hardy-Copson type inequalities and offer new Hardy-Copson type inequalities even for the special cases.

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Some reverse inequalities of Hardy type on time scales

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.

Cited by 16 publications

References 31 publications

A Variety of Nabla Hardy’s Type Inequality on Time Scales

A Variety of Nabla Hardy’s Type Inequality on Time Scales

On some new double dynamic inequalities associated with Leibniz integral rule on time scales

Diamond alpha Hardy-Copson type dynamic inequalities

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