On Hardy–Knopp Type Inequalities with Kernels via Time Scale Calculus

In this paper, we study the inequalities of Hardy–Knopp type with kernel functions which have two nonnegative different weighted functions in two different spaces, in a general domain called a time scale calculus. A time scale calculus T is considered as a unification of the continuous calculus and the discrete calculus. We will prove these inequalities in a time scale calculus to avoid proving them twice once in the continuous case and the second in the discrete case. Also, as special cases of the time scale calculus, we can prove some new inequalities in new different domains. Our results will be proved by using the definition of a general Hardy operator on time scale. These inequalities (when T (cid:31) N ) are essentially new.

In the last decades, a new theory has been discovered to unify the continuous calculus and discrete calculus. It is called a time scale theory. Many authors established dynamic inequalities and generalized them on time scales. For example, see papers [10][11][12][13][14][15][16][17]. In particular, Saker et al. [18] proved that holds. Moreover, the estimate for the constant C > 0 in (12) is where the functions p and q which are nonnegative rdcontinuous functions are called the weighted functions and condition (13) gives the characterization of these two functions which leads to the validation of inequality (12).
In [19], Özkan et al. showed holds for w ∈ C r d ([x, y) T , R) s.t. w(s) ∈ (c, d). Also, they proved that if p ∈ C rd ([y, ∞), R) is a nonnegative function and 2 Journal of Mathematics then the inequality In [19], the authors proved inequality (2) for several functions on time scales and showed that if x ∈ T and w h , h � 1, 2, . . . , n, are nonnegative delta integrable functions, then where Also, they proved that if x, y ∈ T, for h � 1, 2, . . . , n, and assume that p ∈ C rd ([x, y) T , R) and Furthermore, if Ψ: (c, d) ⟶ R is continuous and convex, then In [20],Özkan and Yildirim generalized (16) with kernels and proved that if Ψ: en, the inequality satisfies for all delta integrable functions (25) e aim of this manuscript is to establish some new characterizations for dynamic inequalities of Hardy-Knopp type with kernels in different spaces to prove the following inequality: where 1 < η ≤ q < ∞. e paper is coordinated as follows: in Section 2, we show some basics on T calculus and some theorems needed in Section 3, where we prove the main outcomes. Our key conclusions (when T � R) give the characterizations of inequality (9) proved by Kaijser, Nikolova, Persson, and Wedestig. Also, we give some new characterizations of weights for new general theorems.

Preliminaries
A time scale T is an arbitrary nonempty closed subset of the real numbers R.
e set of all such rdcontinuous functions is denoted by C r d (T, R). To learn more about the time scale calculus, see [21,22]. e derivative of Φϖ and Φ/ϖ (where ϖϖ σ ≠ 0) of two differentiable functions Φ and ϖ is e integration by parts formula on T is e time scales chain rule is such that ϖ: R ⟶ R is continuously differentiable and φ: T ⟶ R is delta differentiable. e Hölder inequality on T is where ζ 0 , ζ ∈ T, λ, φ ∈ C r d (I, R), c > 1, and 1/c + 1/] � 1.

Main Results
In what follows, we define the general Hardy operator A h as follows: is a delta integrable and nonnegative function. roughout this section, we will assume that the functions are nonnegative rd-continuous and the right-hand sides of the inequalities converge if the left sides converge. Now, we are ready to state and prove our main results.

Conclusion and Future Work
In this paper, we present some new inequalities of Hardy-Knopp type with kernel functions which have two nonnegative different weighted functions in two different spaces l η and l q for 1 < η ≤ q < ∞. Also, we established some inequalities using convex functions and applying general Hardy operator on time scale. In the future, we may generalize these results to be with multidimensional time scale. Also, some new dynamic inequalities of Hardy-Knopp type with kernel functions in two different weighted functions in two different spaces l η and l q for η ≤ q < ∞ and 0 < η < 1. We hope general studies about a generalized Hardy operator on time scale and its application for some dynamic inequalities on time scales.

Data Availability
No data were used to support this study.