The weighted discrete dynamic inequalities for 4-convex functions, and its generalization on time scales with constant graininess function

Motivated by Pečarić et al. in (J. Math. Inequal. 11(2):543–5502017), we established here weighted discrete dynamic inequalities for the difference of second divided difference of 4-convex functions. Further we extend and unify the two inequalities, by establishing the theory of n-convexity on time scales having constant graininess function.


Introduction
The two well-known equations of difference and differential equations were born due to the sets of integers and real numbers, respectively. By virtue of a comprehensive study of these kinds of equations, two different mathematical disciplines came into being, known as discrete and continuous analysis. Many productive attempts have been made to bring the two concepts together. Among those, the most significant attempt towards the unification is considered to be the time scale theory. This theory was developed by Aulbach and Hilger [2]. The pioneer of the theory provided efficient tools to unify discrete and continuous problems in his newly introduced theory. The classicist theories on an arbitrary non-empty closed subset T of real numbers were extended to a set designated as time scale. The two well-known particular cases of time scales are the set of integers Z and the set of real numbers R. The applications of this theory have captured the attention of many prolific researchers over the past few years. It has different applications in engineering, biology, social sciences, neutral network, physics, and economics. For additional details and basic notions of time scale calculus, we refer to [3][4][5].
Study and applications of convex functions and their different classes have a significant place in the literature. Here we will discuss a special class of convex functions, known as n-convex functions. The n-convexity or higher order convexity, for n ≥ 3, on an interval was first considered by Hopf [6] in his dissertation. This was further studied in great detail by Popoviciu in [7,8]. A detailed review of this class of functions is given in [9,10]. Convex functions have been defined on time scales by Dinu in 2008 [11]; after that large numbers of inequalities for convex functions on time scales have been developed; see for example [12,13]. To the best of our knowledge n-convexity has not been defined on time scales yet. In [14] Mikić and Pečarić developed some integral inequalities on time scales that hold for n-convex functions but they defined their function on an interval on R. We need to develop n-convex functions on time scales, in order to proceed towards our major results. This paper is organized as follows. In Sect. 2 we give some preliminaries supporting to the central results. Section 3.1 is devoted to defining n-convexity on time scales and establishing a relationship between the nth divided difference and the nth ordered delta derivatives on time scales. We develop discrete dynamic inequalities for the difference of the second divided difference of 4-convex functions in Sect. 3.2. Finally in Sect. 3.3 we unify the inequalities defined in the preceding section, along with their continuous versions established in [1] through the theory of time scales calculus.

Preliminaries
We accumulate in this section some fundamental preliminaries that are used throughout the remaining part of the paper.

Time scale calculus
This part consists of the following basic concepts of time scales related to this article.
We quote [3]: "A time scale is defined to be an arbitrary closed subset of the real numbers R, with the standard inherited topology. Since a time scale may or may not be connected, we need the concept of jump operators, the forward jump operator and the backward jump operator are defined by σ (ι) := inf{s ∈ T : s >ι}, and ρ(ι) := sup{s ∈ T : s <ι}, where inf φ = sup T and sup φ = inf T. Thus, we classify the pointsι ∈ T in such a manner that, if σ (ι) >ι, thenι is right-scattered, and if ρ(ι) <ι, thenι is left-scattered. Points that are right-scattered and left-scattered at the same time are called isolated. Also, if σ (ι) =ι, then ι is said to be right-dense, and if ρ(ι) =ι, thenι is said to be left-dense. Points that are simultaneously right-dense and left-dense are called dense. The mappings μ, ν : are called the forward and backward graininess functions respectively. " Throughout this paper, an interval in time scales is denoted by I T = I ∩ T, where I ⊆ R. We define otherwise.

Definition 2.1 ([3])
Assume u : T − → R is a function and letι ∈ T K . Then we define u (ι) to be the number (provided it exists) with the property that, given any > 0, there is a neighborhood U T ofι such that We call u (ι) the delta derivative of u atι. We say that u is delta differentiable on T K provided u (ι) exists for allι ∈ T K .

Theorem 2.2 ([3])
For allι ∈ T K , we have the following: a. If u is delta differentiable atι, then u is continuous atι. b. If u is continuous atι andι is right-scattered, then u is delta differentiable atι with exists as a finite number. In this case, .

Definition 2.5 ([3]) A function u :
T − → R is called rd-continuous provided it is continuous at right-dense points in T and its left-sided limits exist (finite) at left-dense points in T. The set of rd-continuous functions u : T − → R is denoted by The set consisting of functions u : T − → R which are differentiable and whose derivative is rd-continuous is denoted by
Lemma 2.8 (Rolle's theorem, [5]) If u has at least k ∈ N GZs on T, counting multiplicities, then u has at least k -1 GZs on T K , counting multiplicities.
Remark 2.9 ([3]) u σ and u σ are not equal in general even if both exist which is clear from the given relation although for the time scales with constant graininess functions μ(ι), such as T = R, T = Z, We intensively use this fact to accomplish the key results in Sects. 3.1 and 3.3.

Discrete calculus
This section recalls the following basic lemma from discrete calculus which is of great interest.
Here we use these notations

n-Convex functions
Let u : I − → R, where I is bounded and closed interval contained in R, and let a 1 , a 2 ∈ I then, for all δ ∈ [0, 1], if holds, then u is known as a convex function. Let x 0 < x 1 < · · · < x n be dissimilar points in I. An nth order divided difference of u for these points is defined recursively as We state some basic properties of n-convex functions [9,16]. The 0-convex function is nonnegative function, 1-convex function is simply non-decreasing and a 2-convex function is just a convex. u is n-convex if and only if u (n) ≥ 0, provided that u (n) exists. where n is the nth forward difference.
A 2-convex sequence is just a convex sequence with 2 a n = a n -2a n+1 + a n+2 ≥ 0.
It is clear that, if u : [0, ∞) − → R is a convex function, then a n = u(n) is a convex sequence.

Main results
This section is devoted to providing our main results of this article. We are considering the time scales having constant graininess functions throughout this section.
As there are n adjacent intervals in [t a , t b ] T whose end points are GZs of Ω(s) of order 1, by Rolle's theorem on time scales [5,Lemma 8.9], in consideration of Ω(s) ∈ C n rd ([t a , t b ] T , R), Ω (s) has n GZs of order 1 in interval (3.3). They form at least n -1 intervals, by applying Rolle's theorem again we get n -1 distinct γ i in these intervals on which Ω 2 (s) has GZs.
Thus, there are at least n -1 distinct points in the interval (3.3) at which Ω 2 (s) has GZs.
Thus proceeding we get a point, say γ , in (3.3) as a GZ of the function Ω n (s).
From the above discussion, we immediately conclude to the following result. Then the following estimate is valid: and by applying summation by parts on B, we get note that here we used condition (3.11). Now we have to use the product rule for the forward difference Again by applying the summation by parts formula to the first term, we get Applying the summation by parts formula to the second term this time, we get By applying the summation by parts formula to the second term twice, we get now by breaking up the last term and adding and subtracting the term 1 ι=t a ϑ(ι + 3) ϑ(ι + 3) + ϑ(ι + 4) ϑ(ι + 3) 3 w(ι) ι=t a ϑ(ι + 3) ϑ(ι + 4)ϑ(ι + 3) 3 w(ι).
By using the third term with the difference of (ϑ(ι + 3)) 2 , and the last term with the difference of ϑ(ι + 3) and applying the summation by parts formula to the third term, we get ι=t a ϑ(ι + 3) 2 ϑ(ι + 2) + 2 ϑ(ι + 1) 2 w(ι) We can write ι=t a 2 ϑ(ι + 2) + 2 ϑ(ι + 1) 2 w(ι) Now by using 2-as well as 4-convexity of ϑ(ι) and concavity of the weight function, we get Finally by using the summation by parts formula four times on the first term and two times on the second term, respectively, we get