Generalizations of fractional Hermite-Hadamard-Mercer like inequalities for convex functions

1 Pontificia Universidad Católica del Ecuador, Facultad de Ciencias Naturales y Exactas, Escuela de Ciencias Fı́sicas y Matemáticas, Sede Quito, Ecuador 2 Jiangsu Key Laboratory of NSLSCS, School of Mathematical Sciences, Nanjing Normal University, 210023, China 3 Department of Mathematics, Faculty of Technical Science, University “Ismail Qemali”, 9400 Vlora, Albania 4 Department of Mathematics, Faculty of Science and Arts, Düzce University, Düzce, Turkey

On the one hand, it is well known that RL and GRL fractional integrals have the same importance in theory of integral inequalities, and the GRL fractional integrals are more convenient for calculation.
Therefore it is necessary to study the Hermite-Hadamard integral inequalities by using the GRL fractional integrals while by using the RL fractional integrals. Fortunately, studying the Hermite-Hadamard integral inequalities via the GRL fractional integrations can unify the research of ordinary and fractional integrations. So it is necessary and meaningful to study Hermite-Hadamard integral inequalities via the GRL fractional integrations (see for details [21,[33][34][35][36]).
In this paper, we consider the integral inequality of HHM type that depends on the Hermite-Hadamard and Jensen-Mercer inequalities. For this reason, we recall the Jensen-Mercer inequality: Let 0 < x 1 ≤ x 2 ≤ · · · ≤ x n and α = (α 1 , α 2 , . . . , α n ) nonnegative weights such that n i=1 α i = 1. Then, the Jensen inequality [37,38] is as follows, for a convex function f on the interval [c, d], we have For some results related to Jensen-Mercer inequality, see [39][40][41]. Based on the above observations and discussion, the primary purpose of this article is to establish several inequalities of HHM type for convex functions by using the GRL fractional integrals.

Main Results
Throughout this attempt, we consider the following notations: Proof. From Jensen-Mercer inequality, we have for u, v ∈ [c, d]: Then, for u = tx + (1 − t) y and v = ty + (1 − t) x, it follows that This gives the first inequality in (2.1). To prove the second inequality in (2.1), first we have by the convexity of f : By changing the variables u = tx + (1 − t) y and v = ty + (1 − t) x in (2.5), we have Multiplying both sides of (2.6) by ((y−x)t) t and integrating the result with respect to t over [0, 1], we get which implies that By adding f (c) + f (d) on both sides of (2.7), we can obtain the second inequality in (2.1). Now we give the proof of inequalities (2.2). Since f is convex function, then for all u, v ∈ [c, d], we have for each x, y ∈ [c, d] and t ∈ [0, 1]. Now, by multiplying both sides of (2.9) by ((y−x)t) t and integrating the obtaining inequality with respect to t over [0, 1], we obtain and this completes the proof of the first inequality in (2.2). To prove the second inequality in (2.2), first we use the convexity of f to get Adding (2.10) and (2.11), we get Multiplying both sides of (2.12) by ((y−x)t) t and integrating the result with respect to t over [0, 1], we obtain By using the change of variables of integration and then by multiplying the result by 1 2Λ(1) , we can obtain the second and third inequalities in (2.2). This completes the proof of Theorem 2.1.
Remark 2.1. Let the assumptions of Theorem 2.1 be satisfied. Then, which is derived in [25].
which is derived in [44].
Remark 2.2. If we set x = c and y = d in (2.14), then we have the well-known conformable fractional HH integral inequality: which is derived by Ahmad et al. in [46].
For a convex function f : [c, d] → R, we have the following inequalities for GRL: Proof. From the convexity of f , we have . Thus, the first inequality in (2.17) is proved. To prove the second inequality in (2.17), by using the Jensen-Mercer inequality, we can deduce: By adding (2.20) and (2.21), we obtain By using change of variables of integration and multiplying the result by 1 2∆(1) , we can easily obtain second inequality in (2.17).
• If we put (t) = t Corollary 2.3. For a convex function f : [c, d] → R, we have the following inequalities of HHM type for conformable fractional integrals: (2.23) Proof. By setting (t) = t (y − t) ν−1 in Theorem 2.2, then we have proof of Corollary 2.3.
Remark 2.5. If we set x = c and y = d in (2.23), then we get Corollary 2.4. For a convex function f : [c, d] → R, we have the following inequalities of HHM type for fractional integrals with exponential kernel: Proof. By setting (t) = t ν exp − 1−ν ν t in Theorem 2.2, we get proof of Corollary 2.4. Remark 2.6. If we set x = c and y = d in (2.24), then we get

Related equalities and inequalities
In view of the inequalities (2.1) and (2.17), we can generate some related results in this section.
Proof. By the help of the right hand side of (3.1), we have By applying integration by parts, one can obtain Similarly, one can obtain By making use of S 1 and S 2 in (3.2), we get the identity (3.1).
Remark 3.1. Let the assumptions of Lemma 3.1 be satisfied.
• If x = c and y = d, then Proof. By setting (t) = t (y − t) ν−1 in Lemma 3.1, then we have proof of Corollary 3.1.

Remark 3.2.
By setting x = c and y = d in (3.4), we get Corollary 3.2. Let the assumptions of Lemma 3.1 be satisfied, then the following equality holds for the fractional integrals with exponential kernel: Proof. By setting (t) = t ν exp − 1−ν ν t in Lemma 3.1, we get proof of Corollary 3.2. Remark 3.3. If we set x = c and y = d in (3.5), we get Then, the following equality holds for GRL: Proof. The proof of Lemma 3.2 is similar to Lemma 3.1, so we omit it.
Remark 3.4. Let the assumptions of Lemma 3.2 be satisfied.
• If x = c and y = d, then Lemma 3.2 becomes Corollary 3.3. Let the assumptions of Lemma 3.2 be satisfied, then the following equality holds for the conformable fractional integrals: Proof. By setting (t) = t (y − t) ν−1 in Lemma 3.2, we have proof of Corollary 3.3.
Remark 3.5. If we set x = c and y = d in (3.8), we get Corollary 3.4. Let the assumptions of Lemma 3.2 be satisfied, then the following equality holds for the fractional integrals with exponential kernel: Proof. By setting (t) = t ν exp − 1−ν ν t in Lemma 3.2, we get proof of Corollary 3.4.
Remark 3.6. If we put x = c and y = d in (3.9), we get Then, the following inequality holds for GRL: Proof. In view of Lemma 3.1, we have Then, by using the Jensen-Mercer inequality, we obtain which completes the proof of Theorem 3.1.
• If x = c and y = d, then Theorem 3.1 reduces to [21,Theorem 6]. (3.12) Proof. If we set (t) = t in Theorem 3.1, then we have proof of Corollary 3.5. Corollary 3.6. Let the assumptions of Theorem 3.1 be satisfied. Then, we have the following inequality holds for conformable fractional integrals: (3.13) Proof. By setting (t) = t (y − t) ν−1 in Theorem 3.1, we have proof of Corollary 3.6.
Remark 3.9. If we set x = c and y = d, then we have Corollary 3.7. Let the assumptions of Theorem 3.1 be satisfied. Then, we have the following inequality for fractional integrals with exponential kernel: Proof. By setting (t) = t ν exp − 1−ν ν t in Theorem 3.1, we get proof of Corollary 3.7. Remark 3.10. If we set x = c and y = d in (3.14), then we have  We can apply the Jensen-Mercer inequality due to the convexity of | f | q , to get which completes the proof of Theorem 3.2.

Conclusions
In this work inequalities of Hermite-Hadamard-Mercer type via generalized fractional integrals are obtained. It is also proved that the results in this paper are generalization of the several existing comparable results in literature. As future direction, one may finds some new interesting inequalities through different types of convexities. Our results may stimulate further research in different areas of pure and applied sciences.