Hadamard and Fejér–Hadamard Inequalities for Further Generalized Fractional Integrals Involving Mittag-Leffler Functions

In this paper, generalized versions of Hadamard and Fejér–Hadamard type fractional integral inequalities are obtained. By using generalized fractional integrals containing Mittag-Leffler functions, some well-known results for convex and harmonically convex functions are generalized. The results of this paper are connected with various published fractional integral inequalities.


Introduction
First we give definitions of fractional integral operators which are useful in establishing the results of this paper. In the following, we give fractional integral operators defined by Andrić et al. in [1] via an extended generalized Mittag-Leffler function in their kernels.
e following remark provides connection of Definition 3 with existing fractional integral operators.
e Riemann-Liouville fractional integrals for a func- After introducing generalized fractional integral operators, now we define notions of functions for which generalized fractional integral operators are utilized to get main results of this paper.
Definition 5 (see [10]). Let I be an interval such that I ⊆ R + . en, a function φ: I ⟶ R is said to be harmonically convex, if holds for all a, b ∈ I and t ∈ [0, 1].
In the upcoming section, we give two versions of the Hadamard inequality as well as two versions of the Fejér-Hadamard inequality. eir special cases are also discussed along with noticing connections with published results.

Main Results
First we give the following version of the Hadamard inequality.
(ii) Proof is similar to the proof of (i).

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Now we obtain Fejér-Hadamard type fractional integral inequalities for p-convex function via generalized fractional integral operators; for this, first we prove the following lemma.

Conclusion
We have established Hadamard and Fejér-Hadamard fractional integral inequalities for generalized fractional integrals of p-convex functions. e results of this paper hold simultaneously for convex and harmonically convex functions for different fractional integral operators containing Mittag-Leffler functions.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.