Exploring Generalized Hardy-Type Inequalities via Nabla Calculus on Time Scales

: In this research, we aim to explore generalizations of Hardy-type inequalities using nabla Hölder’s inequality, nabla Jensen’s inequality, chain rule on nabla calculus and leveraging the properties of convex and submultiplicative functions. Nabla calculus on time scales provides a uniﬁed framework that uniﬁes continuous and discrete calculus, making it a powerful tool for studying various mathematical problems on time scales. By utilizing this approach, we seek to extend Hardy-type inequalities beyond their classical continuous or discrete settings to a more general time scale domain. As speciﬁc instances of our discoveries, we have the integral inequalities previously established in the existing literature.

Also, he established if γ > 1 and κ, ψ ≥ 0 s.t.ϕ is convex, then where In the last few years, there has been significant attention given to the study of inequalities ( 1) and ( 2) on a time scale T, which is referred to as an arbitrary nonempty closed subset of R. The emergence of this novel idea has motivated scholars to investigate Hardy-type inequalities on T. The initial exploration of this direction is credited to P. Řehák [16].In fact, he demonstrated that Given that d > 0, h > 1 and κ ≥ 0. If, furthermore, µ(s)/s → 0 as s → ∞, then (h/(h − 1)) h is sharp.Refer to Section 2 for the notations used here and in the upcoming content.
In [17], the authors derived the subsequent Jensen's inequality over T : where is a convex function.
In continuation of this growing trend and with the aim of advancing the exploration of dynamic inequalities over T, we shall present numerous novel of Hardy inequalities within this mathematical framework.The research findings will contribute to the understanding of mathematical inequalities and their applications on time scales, presenting a valuable addition to the existing body of knowledge in this field.Furthermore, these results may have potential implications in various mathematical and applied areas where time scales play a crucial role, such as difference equations, dynamic systems, and more.
While numerous outcomes exist in the realm of T calculus involving the delta derivative, there remains a notable scarcity of research concerning the nabla derivative.As a result, the primary goal of this study is to expand ( 9)- (11) for nabla time scale calculus.The foundational principles of these principal theorems draw inspiration from the work presented in the paper [20], wherein analogous findings were delineated within the domain of time scale delta calculus.
Our discussions are organized as follows.Basic ideas and a number of lemmas related to time scale calculus are presented in the section that follows.The main findings are presented in the final section.

Preliminaries
Before delving into the main results, we will introduce some preliminary concepts and background information to set the foundation for our study.
In the context of any function Γ : T → R, Γ ρ (ς) signifies the value of Γ at ρ(ς).Additionally, we define J T as J T := J ∩ T, J ⊆ R.

Definition 1 ([32]
). Γ is left-dense continuous (ld−continuous), if it exhibits continuity at each left-dense point in T and possesses well-defined right-sided limits at right-dense points within T.
The set encompassing all such ld-continuous functions is denoted as C ld (T, R).
The product κχ is ∇−differentiable at s and holds.

Principal Findings
Throughout our study, we will operate under the assumption that the functions involved are ld-continuous functions and we will consider integrals that exist within the context.Theorem 3. Given β, r ∈ T and considering non-decreasing functions f , ψ ≥ 0, we have Proof.From (23), we find that Let ς ≤ s.Then, f (ς) ≤ f (s) (as f is a nonnegative and nondecreasing), thus As ψ ≥ 0 is a nondecreasing, it follows that Using ( 26) into (24), we get This result corresponds to (22).

Conclusions
In this research, we have successfully demonstrated several dynamic inequalities of Hardy type by utilizing nabla calculus, specifically for convex, submultiplicative functions, and monotone functions.Looking ahead, we intend to extend our exploration by presenting similar inequalities using diamond-α calculus for α ∈ (0, 1) as well as quantum calculus.It is intriguing to consider the prospect of introducing analogous inequalities on time scales through Riemann-Liouville-type fractional integrals.Moreover, there's potential for us to generalize the dynamic inequalities discussed in this article to two or more dimensions, incorporating symmetry in both the functions and variables.The concept of symmetry has various implications for convex functions, submultiplicative functions and Hardy-type inequalities, impacting their properties, behavior, and generalization.Recognizing and utilizing symmetry can aid in proving inequalities and understanding their solutions and applications.By addressing the suggested future research directions, scholars can continue to deepen their understanding of Hardy-type inequalities and their broader implications.This research is a stepping stone towards further advancements in the field.

Author
Contributions: Investigation, software and writing-original draft, H.M.R. and A.I.S.; supervision, writing-review editing and funding, O.S.B. and M.I.M.All authors have read and agreed to the published version of the manuscript.