n-Convexity via Delta-Integral Representation of Divided Difference on Time Scales

We introduce the delta-integral representation of divided difference on arbitrary time scales and utilize it to set criteria for n-convex functions involving delta-derivative on time scales. Consequences of the theory appear in terms of estimates which generalize and extend some important facts in mathematical analysis.


Introduction
Time scale calculus is a well known and rapidly growing theory in mathematical analysis which unifies two distinct well-known mathematical areas named as continuous and discrete analysis. For supplementary details and basics of time scale calculus, we invoke [1][2][3].
The notion of convexity with its various types have a noteworthy presence in literature, see [4][5][6][7] and the references therein. The notion is firstly generalized on an arbitrary time scale in 2008 by Cristian Dinu [8], subsequently a large number of estimation and inequalities for the functions that are convex on time scales are in the continuous state of development, some of them are present in [9,10]. Here we consult with an exclusive variety of these functions, that is n-convex functions. The n-convexity or higher order convexity firstly investigated by Eberhard Hopf [11] in his scholarly thesis. Further it was discussed in different narrations by Popoviciu [12,13]. A comprehensive review of this family of functions is elaborated in [5,14]. In [15] M. Rozarija, and J. Pečarić discussed some "Jensen-Type Inequalities on Time Scales" involving real-valued n-convex functions. Higher order convex functions has been discussed on time scales with constant graininess function by H. A. Baig and N. Ahmad in [16], so there is a need to explore this class of functions on arbitrary time scales.
This article is structured as follows. In section 2 we furnish few preliminaries, utilizing in the main results. Section 3 is dedicated to construct a relationship between nth delta derivatives and nth-order divided difference on arbitrary time scales. Afterward, we presented some mathematical inequalities as consequences of our main results in the last section.

Preliminaries
A time scale T is defined to be an arbitrary closed subset of the real numbers R, with the standard inherited topology. The forward jump operator and the backward jump operator are defined by σ(t) := inf{s ∈ T : s > t}, and ρ(t) := sup{s ∈ T : s < t}, where infφ = supT and supφ = infT. Let u : T → R, u ∆ (t) is representing the first delta derivative of function u at t ∈ T κ . The second-order delta derivative of u at t is defined as, provided it exists Similarly higher-order derivatives are defined as u ∆ n (t) : T κ n → R. The definition for rd-continuous functions can be seen in [2]. The set of rd-continuous functions u : T → R is denoted by The set consisting of first-order delta differentiable functions u and whose derivative is rd-continuous is denoted by The substitution rule and first mean value theorem for delta-integrals in time scales are presented in [1][2][3].
Theorem 2.1. Assume ν : T → R is strictly increasing andT := ν(T) is a time scale. If u ∈ C r d and Then there exists a real number λ satisfying the inequalities m < λ < M such that b a u(t)ν(t)∆t = λ b a ν(t)∆t.
The time scale monomials have been defined in [1,3,17] recursively as g 0 (t, s) = h 0 (t, s) = 1 for s, t ∈ T, These monomials satisfy the following relation for t ∈ T and s ∈ T κ : Remark 2.1. [17] The functions h n and g n satisfy g n (t, s) ≥ 0 and h n (t, s) ≥ 0 for all t ≥ s.
Let us recall the Taylor's formula defined on time scales from [17].
Theorem 2.3. Let u be n-times delta-differentiable on T κ n , t ∈ T and t α ∈ T κ n−1 . We have similarly, higher order convex functions defined on R as well as on Z through nth-order divided difference, in which we randomly select n + 1 points {a 0 , a 1 , . . . , a n } from R or from Z, respectively and compute the nth-order divided difference by the formula [a 0 , a 1 , · · · , a n ; u] = [a 1 , a 2 , · · · , a n ; u] − [a 0 , a 1 , · · · , a n−1 ; u] a n − a 0 . (2.6) If (2.6) is non-negative we say that u is an n-convex function. Here (2.6) remains same for every permutation of n + 1 points.
To construct the criteria for n-convexity we need to introduce the forward operator σ in the definition of higher order convexity. So we adopt the same strategy as we did in [16]. Assume n + 1 distinct points t 0 , · · · , t n ∈ T and arrange them in an increasing order. Relabel these points in the time scalẽ T in terms of forward operator, that is Consequently we can define the nth-order divided difference for n + 1 points as So a function u : T → R, is said to be n-convex if where σ : T T → T T .

Main Results
Here we want to establish a criteria for n-convex function on arbitrary time scales which is stated It is sufficient to prove this onT. Firstly we introduce a new representation of divided difference in terms of delta-integral, that can be seen in the next Theorem.
So that the delta derivative of v (s n ) with respect to s n gives us .
In the next Theorem we establish a relation between nth-order divided difference and nth-delta derivative on arbitrary time scales, since in this result the points t i ∈ T need not to be distinct.
Proof. By using the time scale monomials (2.2) we can write a general notation for the integral Then by the rd-continuity of u ∆ n there exists a λ in this interval that is u ∆ n (ξ) = λ, such that [t 0 , σ(t 0 ), · · · , σ n (t 0 ); u] (h i (s n−i , 0)) = u ∆ n (ξ).
Here, we can directly achieve the next result.
Corollary 3.1. Let u : T → R is n-convex function iff u ∆ n ≥ 0, given that u ∆ n exists.
Another useful property of n-convex function is represented in the next result.

Applications: Inequalities for n-convex functions
Let us present Levinson's type inequality for higher-order convex functions on time scales for this we require the next result. Let Theorem 4.1. Let u is (n + 2)-convex on T. Then for every t ∈ T the function is a convex function.
Proof. By using (3.1), (4.1) can be expressed as Therefore u ∆ n is convex by Theorem 3.3, thus for fixed s j , σ j (t) for j = 1, · · · , n we can write which concludes the proof.
Theorem 4.2. If u is (n + 2)-convex on T, then the given inequality is true Proof. The proof is the direct consequence of Theorem 4.1.
Remark 4.1. Let T = R in Theorem 4.2, inequality (4.2) coincides with inequality (4) in [18], this Levinson's type inequality itself having a great importance in literature which is used to develop further divided difference estimates for n-convex functions in [19].
Further, we present certain useful inequalities involving n-convex functions on time scales by using the criteria for n-convexity, that is u ∆ n ≥ 0.

4)
and if n is even, the given inequality holds Proof. If u is (n + 1)−convex on T κ n which implies that u ∆ n+1 ≥ 0, then u ∆ n is increasing on T κ n , i.e which executes the proof for (4.3).
Let n is odd and t ≤ σ(γ) so that g n−1 (σ(γ), t) ≥ 0, thus we can write which gets the form which executes the proof for (4.4).
Let n is even then we have then by adopting the same steps we can prove (4.5).
Therefore, we can extract the particular cases of Theorem 4.3 by considering different time scales.
First by taking T = R we obtained the following result which agrees Theorem 1 in [20].
. Then for all t ∈ (t α , t β ), the following inequality For odd n the following inequality is true
Theorem 4.5. Let u t : [t α , t β ] → R be an (n + 1)−convex sequence. Then for all t ∈ (t α , t β ), the following inequality holds For odd n the following inequality is true 12) and for even n the following inequality holds (4.14) The next result is obtained by considering n = 1 in (4.3) and (4.4).
, then the given inequalities hold for all The next result is obtained by considering n = 2 in (4.3) and (4.5).
, then the given inequalities hold for all

Conclusion
The notion of n-convexity has been discussed in [16], on specific time scales that are R or hZ.
Here we extend the theory on arbitrary time scale and developed the relationship between the delta derivatives of order n and the nth-order divided difference using integral representation of nth-order divided difference on time scales, see [5,22]. Further we utilized this relationship to derive some dynamic inequalities from which we are able to extract some difference inequalities that are equally important in the study of difference equations and their applications.
Authors Contribution: Both authors have equivalent contribution in this research. Both authors have inspect the manuscript and certified the final version.
Funding Information: The authors acknowledge the moral and financial support by the Higher Education Commission (HEC), Pakistan, through the funding of Indigenous Scholarship phase I, batch V.

Conflicts of Interest:
The author(s) declare that there are no conflicts of interest regarding the publication of this paper.