Some New Generalized Inequalities of Hardy Type Involving Several Functions on Time Scale Nabla Calculus

: In this article, we establish several new generalized Hardy-type inequalities involving several functions on time-scale nabla calculus. Furthermore, we derive some new multidimensional Hardy-type inequalities on time scales nabla calculus. The main results are proved by applying Minkowski’s inequality, Jensen’s inequality and Arithmetic Mean–Geometric Mean inequality. As a special case of our results, when T = R , we obtain reﬁnements of some well-known continuous inequalities and when T = N , the results which are essentially new.

In [13], Özkan et They also proved that if u ∈ C rd ([y, ∞), R) is a non-negative function, and holds for all f ∈ C rd ([y, ∞), R).
In [14], the authors proved the time-scale version of (4) as follows. Let k(λ, θ) ∈ C rd ([r, y) × [r, y), R), u ∈ C rd ([r, y), R) be non-negative functions, f ∈ C rd ([r, y), R), Φ is a continuous and convex function, and Then, where Our aim in this study is to generalize (4) on time-scale nabla calculus of power η ≥ 1 in the form where A, B are positive constants. We will also establish the last inequality for several functions. Furthermore, we will prove the last inequality in multidimensions on time-scales nabla calculus.
The paper proceeds as follows. In Section 2, we state some properties concerning the time-scales nabla calculus needed in Section 3, where we prove the main results. Our main results when T → R, we obtain (4) proved by Kaijser et al. [5] and when T → N, we obtain a new discrete inequality.
In 2008, Ferreira et al. [15] proved Minkowski's inequality on diamond alpha time scales. As a special case of this inequality (when α = 0), we get Minkowski's inequality on time-scale nabla calculus as follows.
, c, d ∈ R be ld-continuous and Φ be continuous and convex. Then, If Φ is a concave function, then (13) will be reversed.

Main Results
Throughout this section, we will assume that the functions (without mention) are non-negative ld-continuous functions and the integrals in the statements of the theorems are convergent. We define the general Hardy operator A as follows where λ > r and ∈ C ld ([r, y] T , R + ) and (λ, ϑ) ∈ C ld ([r, y] T × [r, y] T , R + ). Now, we state and prove our main results.
Theorem 4. Let r, y ∈ T, η ≥ 1 and κ, ω be weighted functions, such that Furthermore, assume that χ, ξ defined on (c, d), −∞ < c < d < ∞ and ξ is a convex function, such that where A, B are positive constants; then holds for the non-negative function .
The following theorem is proved for several functions.

Conclusions
In this research, we generalize some new inequalities on time-scale nabla calculus. We will also establish some dynamic inequalities for several functions. Furthermore, we will establish these inequalities in multiple dimensions on time-scales nabla calculus. All of these inequalities can be proved by applying Minkowski's inequality, Jensen's inequality and Arithmetic Mean-Geometric Mean inequality. In the future, we hope to study these dynamic inequalities via conformable nabla fractional calculus on time scales.