New generalized fractional versions of Hadamard and Fejér inequalities for harmonically convex functions

The aim of this paper is to establish new generalized fractional versions of the Hadamard and the Fejér–Hadamard integral inequalities for harmonically convex functions. Fractional integral operators involving an extended generalized Mittag-Leffler function which are further generalized via a monotone increasing function are utilized to get these generalized fractional versions. The results of this paper give several consequent fractional inequalities for harmonically convex functions for known fractional integral operators deducible from utilized generalized fractional integral operators.

We are interested in utilizing fractional integral operators (1.9), (1.10) for the establishment of Hadamard and Fejér-Hadamard inequalities for harmonically convex functions. The classical Hadamard inequality is an elegant geometric interpretation of convex functions.
Hadamard inequality is stated in the following theorem: , be a convex function. Then the following inequality holds: (1.11) Fejér-Hadamard inequality is a weighted version of Hadamard inequality proved by Fejér in [11] which is stated in the following theorem: Next we give the definition of harmonically convex functions [14].

Definition 6
Let I be an interval of nonzero real numbers. Then a function f : I → R is said to be harmonically convex if holds for all a, b ∈ I and t ∈ [0, 1]. If the reversed inequality holds in (1.13), then f is called a harmonically concave function.
In Sect. 3, we prove two fractional versions of Hadamard and two fractional versions of Fejér-Hadamard-type inequalities for harmonically convex functions by using fractional integral operators (1.9) and (1.10). Furthermore, the associated published results are obtained which are identified in remarks, some corollaries are also given.

Main results
, and g is differentiable and strictly increasing. If f is a harmonically convex function on [a, b], then for fractional integral operators (1.9) and (1.10) we have ; p Proof Since f is harmonically convex on [a, b], for x, y ∈ [a, b], the following inequality holds: . dt dt. (2.4) and in (2.4) and using (1.9), (1.10), the first inequality of (2.1) can be obtained. On the other hand, using harmonic convexity of f , we have ; p) and then integrating over [0, 1], we get dt and in (2.6), and using (1.9), (1.10), the second inequality of (2.1) can be obtained.

Corollary 2.3
If we take ψ(x) = x in Theorem 2.1, then we get the following inequalities: where g is the reciprocal function.
The following lemma is useful to give the next result.
, be functions such that f is positive, f ∈ L 1 [a, b], and g is differentiable and strictly increasing.

Corollary 2.12
Setting ω = p = 0 and g = I in Theorem 2.10, we get the following inequalities via Riemann-Liouville fractional integrals: