Unifications of Continuous and Discrete Fractional Inequalities of the Hermite–Hadamard–Jensen–Mercer Type via Majorization

The main objective of the paper is to develop an innovative idea of bringing continuous and discrete inequalities into a uniﬁed form. The desired objective is thus obtained by embedding majorization theory with the existing notion of continuous inequalities. These notions are applied to the latest generalized form of the inequalities, popularly known as the Hermite–Hadamard–Jensen–Mercer inequalities. Moreover, the frequently-used Caputo fractional operators are employed, which are rightly considered critical, especially for applied problems. Both weighted and unweighted forms of the developed results are discussed. In addition to this, some bounds are also provided for the absolute diﬀerence between the left- and right-sides of the main results.


Introduction
e field of mathematical inequalities contributes to a wide area of research in mathematics. With the passage of time, this field has emerged as a separate discipline, despite the fact that it was being used as a tool earlier. e addition of the notion of convexity enriched its literature and stimulated a new trend among researchers. As a result, many new inequalities came to the surface. ese inequalities are (but not limited to) Ostrowski inequalities [1], Jensen's inequalities [2], the Jensen-Mercer inequalities [3], Fejér inequalities [4], Hermite-Hadamard inequalities [5], and their various variants. e Hermite-Hadamard inequality is believed to be the most widespread inequality in the literature and has received much attention in the last few years. is inequality is defined as follows: If ϕ: I ⟶ R is a convex function with ϑ, θ ∈ I such that ϑ < θ then e direction of the inequality given in (1) reverses whenever the function ϕ is concave. is inequality has been established for different generalized convex functions, for example, s− convex [6], η− convex [7], strongly convex [8], and coordinate convex function [9]. Research works in this field have also been extended to the theory of fractional calculus. As there are multiple numbers of fractional operators but due to our interest, we limit ourselves to the wellknown Caputo fractional operators. eir definition is given as follows: Definition 1 (Caputo fractional derivative operators). Consider a function ϕ ∈ C n [ϑ, θ] (the space of functions whose n th -derivative exist and continuous on Where c D α ϑ + ϕ(z) and c D α θ − ϕ(z) stand for the left-and right-sided Caputo fractional derivative operators, respectively.
Niezgoda [22] has used the concept of majorization and extended the Jensen-Mercer inequality given as follows: Theorem 2 (Majorized discrete Jensen-Mercer inequality). Let us consider a convex function ϕ defined on the interval I, r × l real matrix (x is ) , and l− tuple δ � (δ 1 , . . . , δ l ) such that δ s , x is ∈ I for all i � 1, 2, . . . , r , s ∈ 1, . . . , l { } with σ i ≥ 0, r i�1 σ i � 1. If δ majorizes every row of (x is ), then we have e following lemmas will help us to prove our main results [23]. Lemma 1. Let us consider a convex function ϕ defined on the interval I, r × l real matrix (x is ), and two l− tuples δ � (δ 1 , . . . , δ l ), p � (p 1 , . . . , p l ), such that δ s , Lemma 2. Let us consider a convex function ϕ defined on the interval I, r × l real matrix (x is ), and two l− tuples δ � (δ 1 , . . . , δ l ), p � (p 1 , . . . , p l ), such that δ s , x is ∈ I, σ i ≥ 0, r i�1 σ i � 1, p s ≥ 0, with p l ≠ 0, η � 1/p l , for all i � 1, 2, . . . , r, s ∈ 1, . . . , l { }. If for each i � 1, . . . , r, (δ s − x is ) and x is are monotonically in the same sense and l s�1 p s δ s � l s�1 p s x is , e theory of majorization has been successful in drawing the attention of researchers working in various fields. It has been used as a key element in solving complicated optimization problems [24,25]. Some more recent applications of majorization theory in signal processing and communication can be seen in [26,27]. For further successive work carried out via the concept of majorization, one is referred to [28][29][30][31][32][33][34] and the references therein. In the present era, despite the existence of various diverse research fields, the shrinking of more than one research field into one is direly needed. e reason is that new ideas grow fast when they attract the attention of a maximum number of researchers. In our case, since inequalities have two main aspects, one is that of continuous inequalities and the other is of discrete inequalities. Both subfields have been absorbing the attention of many researchers at the same time. e fault is that the majority of the results are based only on simple conversions from discrete to continuous or vice versa. e concept of adding new ideas or strengthening an existing one is rarely utilized. In such a situation, there is a need for the provision of such a platform which can play the role of bringing researchers from the abovementioned subfields together and utilize their energies and efforts in one direction. e present attempt may be considered one of the endeavors in this regard. e present paper is summarized as follows: first of all, eorem 3 is devoted to the establishment of a new unified form of Hermite-Hadamard-Jensen-Mercer inequality. is objective is achieved by utilizing the majorized l− tuples in the context of Caputo fractional operators. A slightly different variant of eorem 3 is presented in the form of eorem 4. In order to verify and provide proof of the fact that the newly-obtained results are the unifications and generalizations of those already existing results, Remark 1 and Remark 2 are presented. In addition to this, weighted versions of the obtained results are also provided, taking the weighted generalized Mercer's inequality into account.
ese weighted results can be traced to eorem 5 and eorem 6. Moreover, two new identities, connected with the right-and left-sides of eorem 3 and eorem 4, respectively, are discovered. Employing these lemmas, various bounds associated with the absolute difference of the two right-and left-most terms in the main results are obtained.
ese results are discussed in eorem 7, eorem 8, eorem 9, eorem 10, and eorem 11. Remark 5, Remark 6, and Remark 7 show that the newly-derived identities also generalize those previously-defined identities, while Remark 8 discusses the previous version of eorem 10. Corollary 1 gives details about a previous bound while Corollary 2, and Corollary 3 provide information about the classical integral versions of eorem 9 and eorem 11. At the end, conclusion of the overall attempt is presented.

Journal of Function Spaces 3
Proof. It can be written as Since ϕ (n) is a convex function, therefore (12) gives the following inequality: By multiplying both sides of (13) by t n− α− 1 and then integrating over t ∈ [0, 1], we get In order to apply the definition of the Caputo fractional operators in (14), first, we show that By the hypotheses, we have x ≺ δ and y ≺ δ, therefore Also, 4 Journal of Function Spaces By substituting (17) in (16), and adding l s�1 δ s to both sides, we get Now (14) implies and so us, the first inequality of (11) is completed. Now, using the convexity of ϕ (n) , we obtain the second inequality in the following manner: Adding (21) and (22), and then applying eorem 2 for r � 1 and σ 1 � 1, we obtain By multiplying both sides of (23) by t n− α− 1 and then integrating over t ∈ [0, 1], we get the second and third inequality in (11). □ Remark 1. For the hypothesis of eorem 3, if l � 2, then we get the following inequality: Moreover, for n � 1 and α � 0, we obtain the result of Kian and Moslehian [35].
Another result for the Hermite-Hadamard-Jensen-Mercer fractional inequality is given as follows: Proof. Let us consider t ∈ [0, 1]. To prove the required result, we proceed as follows: Since ϕ (n) is a convex function, therefore (26) gives the following inequality: By multiplying both sides of (27) by t n− α− 1 and then integrating over t ∈ [0, 1], we obtain Following the same procedure, as given in the proof of eorem 3, we can show that Now, from (28), we deduce So, we have Journal of Function Spaces is proves the first inequality in (25).

□
We establish the following result for the Caputo fractional operators on the basis of Lemma 1. 8

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Proof. It can be written as Since ϕ (n) is a convex function, therefore (37) gives the following inequality: By multiplying both sides of (38) by t n− α− 1 and then integrating over t ∈ [0, 1], we get In order to apply the definition of the Caputo fractional operators in (39), first, we show that Also, x l > y l ⇒ p l x l > p l y l ⇒ p l x l − p l y l > 0. (43) and so us, we achieved the first inequality of (36).
To prove the second inequality, from the convexity of ϕ (n) we may write that Adding (46) and (47) and then using Lemma 1 for r � 2, σ 1 � t, and σ 2 � 1 − t, we obtain By multiplying both sides of (48) by t n− α− 1 and then integrating over t ∈ [0, 1], we get the second and third inequality in (36).

Remark 4.
e weighted versions of eorem 4 can be obtained in a similar fashion.

Bounds Associated with the Main Results
In this section first, we discover two new identities associated with the right-and left-sides of the main results. en utilizing these identities, we establish bounds for the absolute difference of the two right-and left-most terms of the main results.

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Proof. To prove our required result, we consider that (52) Assuming that l s�1 δ s − l− 1 s�1 y s < l s�1 δ s − l− 1 s�1 x s and using integration by parts formula, we obtain Similarly, 12

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Proof. From Lemma 3, it follows that Using eorem 2 for r � 2, σ 1 � t, and σ 2 � 1 − t in (59) as a consequence of the convexity of |ϕ (n+1) |, we obtain Now finding C 1 and C 2 , we have Journal of Function Spaces Adding C 1 and C 2 , we get Inserting (62) in (60), we achieve (58).