Fractional Hermite-Hadamard inequalities containing generalized Mittag-Leffler function

The objective of this paper is to establish some new refinements of fractional Hermite-Hadamard inequalities via a harmonically convex function with a kernel containing the generalized Mittag-Leffler function.


Introduction
A function f : I → R is said to be convex if f (t)x + ty ≤ (t)f (x) + tf (y), ∀x, y ∈ I, t ∈ [, ].
Convexity plays a pivotal role in different fields of pure and applied sciences. Another fact that makes it more attractive is its close relationship with theory of inequalities. In particular, integral inequalities have been obtained via convex functions. Inspired by the research work in this field, many authors introduced new extensions of classical convex functions; see, for example, [-] and the references therein. Recently, Işcan [] introduced and investigated the notion of harmonically convex functions. These days the class of harmonically convex functions is receiving much attention by many researchers. For more details, see [-]. Hermite and Hadamard independently obtained an integral inequality that provides us a necessary and sufficient condition for a function to be convex. This famous result reads as follows.
Let f : I ⊇ [a, b] → R be a convex function, then The main motivation of this paper is to obtain new refinements of fractional Hermite-Hadamard-type inequalities via harmonically convex functions in connection with the generalized Mittag-Leffler function, which even generalizes the classical Riemann-Liouville fractional integral operators. We also discuss some particular cases.

Preliminaries
In this section, we discuss some preliminary concepts and facts. Recently, Işcan [] obtained several inequalities of Hermite-Hadamard type via harmonic convex functions. The class of harmonic convex functions is defined as follows.
for all x, y ∈ I and t ∈ [, ]. If (.) holds in the reversed sense, then f is said to be harmonic concave.

Results and discussions
In this section, we discuss our main results. We write Proof Since g is harmonically symmetric with respect to ab b-a , using Definition ., we have Hence, in the following integral, setting u =  a +  bt and du = -dt gives This completes the proof.
→ R is integrable and harmonically symmetric with respect to ab a+b , then the following equality holds: Proof It suffices to show that By Lemma ., integrating by parts, we have Thus Analogously, Also,
Remark In Theorem ., if we take ω = , then we get the known inequality of Işan et al. [] If g : [a, b] → R is nonnegative, integrable, and harmonically symmetric with respect to ab a+b , then the following inequality holds: Proof Since f ia a harmonically convex function on [a, b], multiplying both sides of (.) by t ν- E γ ,δ,k μ,ν,l (ωt μ )g( ab t  a+ -t  b ) and then integrating the resulting inequality over [, ], we obtain dt.
Since g is harmonically symmetric with respect to ab a+b , using Definition ., we have g( Hence, using Lemma ., we obtain For the proof of the second inequality in (.), we first note that f is a harmonically convex function. Then, multiplying both sides of (.) by t ν- E γ ,δ,k μ,ν,l (ωt μ )g( ab t  a+ -t  b ) and integrat-ing the resulting inequality over [, ], we obtain dt.
From this, using Lemma ., we get From (.) and (.) we obtain (.). The proof is completed.
where a, b ∈ I, a < b. If |f | is a harmonically convex function and g : [a, b] → R is continuous and harmonically symmetric with respect to ab a+b , then the following inequality holds: