New “Conticrete” Hermite–Hadamard–Jensen–Mercer Fractional Inequalities

: The theory of symmetry has a signiﬁcant inﬂuence in many research areas of mathematics. The class of symmetric functions has wide connections with other classes of functions. Among these, one is the class of convex functions, which has deep relations with the concept of symmetry. In recent years, the Schur convexity, convex geometry, probability theory on convex sets, and Schur geometric and harmonic convexities of various symmetric functions have been extensively studied topics of research in inequalities. The present attempt provides novel portmanteauHermite–Hadamard– Jensen–Mercer-type inequalities for convex functions that unify continuous and discrete versions into single forms. They come as a result of using Riemann–Liouville fractional operators with the joint implementations of the notions of majorization theory and convex functions. The obtained inequalities are in compact forms, containing both weighted and unweighted results, where by ﬁxing the parameters, new and old versions of the discrete and continuous inequalities are obtained. Moreover, some new identities are discovered, upon employing which, the bounds for the absolute difference of the two left-most and right-most sides of the main results are established.


Introduction
Mathematical inequalities have successfully extended their influence to various fields of science and engineering, and they are now accepted and taught as some of the most applicable disciplines of mathematics. Their fruitful applications can be found in, but not limited to, areas such as information theory, economics, engineering, and biology [1,2]. On the basis of such applicability, inequalities and their associated theory have been developed rapidly, where various new and generalized forms of them have come to the surface. For instance, the Hermite-Hadamard inequality [3], Jensen's inequality [4], the Jensen-Mercer inequality [5], the Ostrowski inequality [6], and the Fejér inequality [7] are some names that are immensely popular with researchers. In the present age, researchers are particularly taking interest in generalized inequalities containing various of the above-mentioned versions in one form. In this regard, the Fejér inequality, the Jensen-Mercer inequality, and the Hermite-Jensen-Mercer inequality are commonly known. We selected the most generalized and latest one, that is the Hermite-Jensen-Mercer inequality. This double inequality has recently attracted researchers' attention because it unifies the remarkable Hermite-Hadamard, Jensen, and Mercer inequalities. The fractional Hermite-Jensen-Mercer inequality is stated as follows: Let f be the real-valued convex function defined on the interval [δ 1 , δ 2 ] of real numbers and [x 1 , y 1 ] ⊂ [δ 1 , δ 2 ], with α > 0. The Hermite-Jensen-Mercer inequality is given as [8]: where J α x + 1 and J α y − 1 respectively represent the left-and right-sided Riemann-Liouville integrals of fractional order α defined as follows: In the present study, one of the reasons for the selection of Riemann-Liouville operators is that these operators have some advantages as compared to other fractional operators. For example, the Riemann-Liouville fractional operators do not need the function to be continuous or differentiable at the origin. In addition to this, these operators can be used for the best descriptions and modeling of phenomena having power-law behaviors because they contain a power function as a kernel in their integral transforms. However, the related research can also be conducted for other fractional operators, such as that of Caputo's, Hadamard's, Katugampola, or generalized k-fractional operators.
As mentioned above, there is a growing trend among researchers to combine different research fields into one. In this regard, it is better to develop such ideas that bring researchers of related fields together. In the field of inequalities, up to now, there are two main concepts (which are continuous and discrete) where mathematicians are conducting research independently. In both cases, researchers have been developing generalized or unified inequalities using (sometimes) generalized integral operators and sometimes a generalized type of convexity, or sometimes, they use both [24]. As a result, they provide a unique platform to researchers working with different integrals or convex functions. In this stage, there is a necessary notion whose applications can lead us to the inequalities that are a mixture or combination of both discrete and continuous versions. The theory of majorization is one of these that fulfills these criteria. The present attempt may be considered as one of the fruitful endeavors in this direction.
Moreover, the name "conticrete" is assigned on the basis of one of the English language rules for "blending or coining of words", according to which "brunch" is used for the meal taken in between "breakfast" and "lunch". Similarly the word "smog" has been created by blending the two words "smoke" and "fog". Here in our case, the word conticrete means mixture of continuous and discrete inequalities.
The main results of the present paper are organized as follows: In Theorem 2, the generalized fractional portmanteauform of the Hermite-Hadamard-Jensen-Mercer-type inequalities is obtained using Riemann-Liouville fractional integrals. Remarks 1 and 2 show that these inequalities cover those previously presented versions of fractional Hermite-Hadamard-Jensen-Mercer-type inequalities, and they also unify continuous and discrete inequalities of the Hermite-Hadamard-, Jensen-, and Mercer-types. In Theorem 3, another form of the Hermite-Hadamard-Jensen-Mercer-type inequality for fractional integrals is developed. Theorems 4 and 5 present weighted versions of the obtained results. Lemmas 3 and 4 contain new identities associated with the right side of Theorem 2 and with the left side of Theorem 3, respectively. Theorems 6 and 7 are proven on the basis of Lemma 3, and they present various bounds for the absolute difference of the two rightmost terms in Theorem 2. Theorems 8-10 are proven on the basis of Lemma 4, and they present various bounds for the absolute difference of the two left-most terms in Theorem 3. Corollaries 1-5 provide information about the classical integral forms of the main obtained results. At the end, the conclusion of the whole research work is presented.

Main Results
In the underlying theorem, we deduce the Hermite-Hadamard inequality of the Jensen-Mercer-type for fractional integrals. Theorem 2. Let δ = (δ 1 , . . . , δ l ), x = (x 1 , . . . , x l ), and y = (y 1 , . . . , y l ) be three l-tuples such that δ s , x s , y s ∈ I, for all s ∈ {1, · · · , l}, x l > y l , α > 0 and f be a convex function defined on I. If δ majorizes both x and y, then: Proof. We may write: Using the convexity of f in (4), we have: By multiplication of t α−1 with (5) on both sides and taking integration with respect to t, we obtain: To apply the definition of the fractional integral in (6), first we show that: As: Furthermore, x ≺ δ and y ≺ δ. Then, we may write: Using (7) in (8), we obtain: Adding l ∑ s=1 δ s to both sides of (9), we deduce: and so: Thus, the first inequality of (3) is complete. To achieve the second inequality, we utilize the convexity of f as follows: Adding (11) and (12), then applying Theorem 1 for n = 1 and σ 1 = 1, we obtain: By multiplication of t α−1 with (13) on both sides and taking integration with respect to t, we acquire the second and third inequality in (3).

Remark 1.
Taking the same hypothesis, Theorem 2 gives the underlying inequality for the case of α = 1.

Remark 2.
Theorem 2 gives the following inequality for l = 2, which was proven by Öagülmüş and Sarikaya in [8].
Adopting the same procedure, we give another Hermite-Hadamard inequality of the Jensen-Mercer-type for fractional integrals as follows.
Theorem 3. Let all conditions in the hypothesis of Theorem 2 hold true, then: Proof. For t ∈ [0, 1], it may be written: Using the convexity of f in (15), we have: By multiplication of t α−1 with (16) on both sides and taking integration with respect to t, we obtain: In a similar manner as adopted in Theorem 2, we can show that: Now, (17) implies: Therefore, we have: This proves the first inequality in (14).
With the purpose of proving the second inequality of (14), we use Theorem 1 for n = 2, σ 1 = t 2 and σ 2 = 2−t 2 as follows: and: Adding (18) and (19), we obtain: By multiplication of t α−1 with (20) on both sides and taking integration with respect to t, we acquire the second inequality of (14).

Remark 3.
For the case of l = 2, the inequality (14) reduces to the following inequality proven by Öagülmüş and Sarikaya in [8].
The underlying theorem includes a result based on Lemma 1.
Proof. We may write: Using the convexity of f in (22), we have: By multiplication of t α−1 with (23) on both sides and taking integration with respect to t, we obtain: In order to apply the definition of the fractional integral in (24), first we show that: As: x l > y l ⇒ p l x l > p l y l ⇒ p l x l − p l y l > 0.
Using (25) in (26), we obtain: Adding l ∑ s=1 η p s δ s to both sides of (27), we deduce: and so: Thus, we achieve the first part of (21). To achieve the second part, from the convexity of f , we may write: and: Adding (29) and (30) and then using Lemma 1 for n = 2, σ 1 = t, and σ 2 = 1 − t, we obtain: By multiplication of t α−1 with (31) on both sides and taking integration with respect to t, we obtain the second and third part of (21).
The underlying theorem includes a result based on Lemma 2. (32) Proof. In similar manner as adopted in Theorem 4, we can easily obtain (32) using Lemma 2.

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. Then, utilizing these identities, we establish bounds for the absolute difference of the two right-most and left-most terms of the main results. Lemma 3. Let δ = (δ 1 , . . . , δ l ), x = (x 1 , . . . , x l ), and y = (y 1 , . . . , y l ) be three l-tuples such that δ s , x s , y s ∈ I, for all s ∈ {1, · · · , l}, α > 0, t ∈ [0, 1] and f be a differentiable function defined on I. If f ∈ L(I), then: Proof. To prove our required result, we consider that: Assume that x s , and using integration by parts formula, we obtain: Likewise, Now, we have: Multiplying both sides by , we obtain (33).
On the basis of Lemma 3, we give the following results.
For some further results, we establish another lemma as follows.

Lemma 4.
Let all conditions in the hypothesis of Lemma 3 hold true, then: Proof. Adopting the same procedure as given in the proof of Lemma 3, it can be easily proven.

Remark 9.
For the selection of l = 2, the equality (42) reduces to the equality (3.3), which was proven in [8].
Now, we give some results on the basis of Lemma 4, as: Theorem 8. Let all conditions in the hypothesis of Theorem 6 hold true, then: By utilizing Theorem 1 for n = 2, σ 1 = 2−t 2 , and σ 2 = t 2 in (44), we obtain: Hence, the proof is accomplished.