Some dynamic Hardy-type inequalities with negative parameters on time scales nabla calculus

: In this paper, we establish some new dynamic Hardy-type inequalities with negative parameters on time scales nabla calculus by applying the reverse H¨older’s inequality, integration by parts, and chain rule on time scales nabla calculus. As special cases of our results (when T = R ), we get the continuous analouges of inequalities proven by Benaissa and Sarikaya, and when T = N 0 , the results to the best of the authors’ knowledge are essentially new.

In 1927, Hardy and Littlewood [3] proved that if 0 < < 1, (ξ) ≥ 0 for ξ ∈ (0, ∞), and where the constant ( / (1 − )) is the best possible.The Hardy inequalities mentioned above are proved for a positive parameter 0 < < 1 and > 1.Also, these inequalities depend on the power rule inequality in case of the positive parameter.So, some authors discovered new inequalities of Hardy type with negative parameters and noted that they proved these inequalities with a different technique which depends on the power rule inequality.
The objective of this paper is to introduce novel generalizations of the continuous the inequalities (1.5)-(1.7)on nabla time scales.The proofs of these results rely on employing the reverse Hölder's inequality and the chain rule formula adapted to time scales.
This paper is divided into three sections: In Section 2, we present some lemmas on time scales needed in Section 3 where we prove our results.These results as special cases when T = R give the inequalities (1.5)-(1.7),while for T = N 0 , the results are fundamentally original.

Preliminaries and basic lemmas
In 2001, Bohner and Peterson [27] introduced the time scale T as an arbitrary nonempty closed subset of the real numbers R. Also, they defined the backward jump operator by ρ(τ) := sup{s ∈ T : s < τ}.For any function : Definition 2.1.[28] A function λ : T → R is left-dense continuous or ld-continuous provided that it is continuous at left-dense points in T and its right-sided limits exist at right-dense points in T. The space of ld-continuous functions is denoted by C ld (T, R).
The set T κ is derived from the time scale T as follows: If T has a left-scattered maximum m, then T κ = T − {m} .Otherwise, T κ = T.In summary, Definition 2.2.[28] A function ψ : T → R is said to be ∇-differentiable at ϑ ∈ T κ if ψ is defined in a neighbourhood U of ϑ and there exists a unique real number ψ ∇ (ϑ), called the nabla derivative of ψ at ϑ, such that for each > 0, there exists a neighbourhood N of ϑ with N ⊆ U and Theorem 2.1.[28] Assume ψ, Θ : T → R are nabla differentiable at ϑ ∈ T. Then: (1) The product ψΘ : T → R is nabla differentiable at ϑ, and we get the product rule (2) If Θ(ϑ)Θ ρ (ϑ) 0, then ψ/Θ is nabla differentiable at ϑ, and we get the quotient rule Lemma 2.1.[29] Let ψ : R → R be continuously differentiable and suppose that Θ : T → R is continuous and nabla differentiable.Then, ψ • Θ : T → R is nabla differentiable and We then define the nabla integral of ψ by ϑ ψ(s)∇s = Λ(ϑ) − Λ( ), ∀ϑ ∈ T.

Conclusions
In this paper, we established some new generalizations of the continuous inequalities on nabla calculus time scales.These inequalities were proved by employing the reverse Hölder's inequality and the chain rule formula adapted to time scales.

Use of AI tools declaration
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.