Extensions of Hermite–Hadamard inequalities for harmonically convex functions via generalized fractional integrals

In the paper, the authors establish some new Hermite–Hadamard type inequalities for harmonically convex functions via generalized fractional integrals. Moreover, the authors prove extensions of the Hermite–Hadamard inequality for harmonically convex functions via generalized fractional integrals without using the harmonic convexity property for the functions. The results offered here are the refinements of the existing results for harmonically convex functions.


Introduction
The Hermite-Hadamard inequality, which is the first basic result of convex mappings with a natural geometric interpretation and extensive use, has attracted attention with great interest in elementary mathematics. Many mathematicians have devoted their efforts to standardization, refining, imitation, and expansion into various categories of works such as convex mappings.
Inequalities found by C. Hermite and J. Hadamard for convex mappings are very important in literature (see [1]). These inequalities state that if F : I → R is a convex function on the interval I of real numbers and κ 1 , κ 2 ∈ I with κ 1 < κ 2 , then (1.1) Both inequalities hold in the reversed direction if F is concave. For further studies of this area, one can consult . For brevity, in the upcoming results, we use the subsequent notations: Mappings , * , , * : [0, 1] → R are defined by and Now we give the definition of the generalized fractional integrals (GFIs) given by Sarikaya and Ertuğral in [23].

Definition 1
The left-sided and right-sided GFIs are denoted by κ 1 + I ϕ and κ 2 -I ϕ and defined as follows: where a function ϕ : [0, ∞) → [0, ∞) satisfies the condition 1 0 Recently, the authors gave some refinements of Hermite-Hadamard inequalities for GFIs under the condition of convexity, as follows.
In [27], İşcan and Wu gave the inequalities of Hermite-Hadamard type for harmonically convex functions via Riemann-Liouville fractional integrals.
If F is a harmonically convex function on [κ 1 , κ 2 ], the following double inequality holds for the Riemann-Liouville fractional integrals: where α > 0.
In [28], Zhao et al. gave the following Hermite-Hadamard type inequalities for harmonically convex functions by utilizing GFIs.

Theorem 6
Let F : I ⊆ (0, +∞) → R be a mapping such that F ∈ L([κ 1 , κ 2 ]). If F is a harmonically convex mapping on [κ 1 , κ 2 ], then the following inequalities hold for the GFIs: (1.10) In [29], F. Chen gave the following useful lemma and the lower and upper bounds of the left-and right-hand sides of inequalities (1.9) as follows.

Lemma 1 A mapping
, then the following inequalities hold for the Riemann-Liouville fractional integrals: In [30], Budak et al. gave the following inequalities for harmonically convex mappings.

Hermite-Hadamard type inequalities
In this portion, we deal with some new inequalities of Hermite-Hadamard type for harmonically convex mappings by applying GFIs.

Corollary 2 Under the assumptions of Theorem
, then we have the following inequality for the k-Riemann-Liouville fractional integrals: Theorem 11 Let F : I ⊆ (0, +∞) → R be a function such that F ∈ L([κ 1 , κ 2 ]). If F is a harmonically convex function on [κ 1 , κ 2 ], then the following inequalities hold for the GFIs: Proof Since F is a harmonically convex function on [κ 1 , κ 2 ], we have both sides of inequality (2.8) and integrating the resultant one with respect to τ over [0, 1], we obtain By Hence we have the first inequality in (2.7).
To prove the second inequality in (2.7), first we note that F is a harmonically convex function, we get Adding (2.9) and (2.10), we have both sides of inequality (2.11) and integrating the resultant one with respect to τ over [0, 1], we obtain By changing the variable of integration, we have the second inequality in (2.7).

Corollary 3
Under the assumptions of Theorem 11, if we set ϕ(τ ) = τ α (α) , then we have the following inequalities for the Riemann-Liouville fractional integrals: Corollary 4 Under the assumptions of Theorem 11, if we put ϕ(τ ) = τ α k k k (α) , then we have the following inequalities for the k-Riemann-Liouville fractional integrals:

Extension of Hermite-Hadamard type inequalities
In this section, we give the following inequalities which give the above and below bounds for the left-and right-hand sides of inequalities (2.1) and (2.7). We prove inequalities (2.1) and (2.7) under the condition φ (κ 1 + κ 2x) ≥ φ (x) instead of the harmonic convexity of F .

Corollary 6
Under the assumptions of Theorem 12, if we put ϕ(τ ) = τ α k k k (α) , then we have the following inequalities for the k-Riemann-Liouville fractional integrals: .
Remark 4 Under the assumptions of Theorem 13, if we put ϕ(τ ) = τ , then we have the following inequalities: Corollary 7 Under the assumptions of Theorem 13, if we set ϕ(τ ) = τ α (α) , then we have the following inequalities for the Riemann-Liouville fractional integrals:
, then the following inequalities hold for the GFIs: Proof By using the change of variables, we have By equality (3.16), we get Using the fact that We also have By using equality (3.19) and the assumption m < φ (u) < M, u ∈ [κ 1 , κ 2 ], we obtain Integrating inequality (3.20) with respect to τ on [x, κ 1 +κ 2 2 ], we get By equality (3.18), we have (3.
That is, we get which gives inequality (3.15).

Corollary 9
Under the assumptions of Theorem 15, if we set ϕ(τ ) = τ α (α) , then we have the following inequalities for the Riemann-Liouville fractional integrals: Corollary 10 Under the assumptions of Theorem 15, if we put ϕ(τ ) = τ α k k k (α) , then we have the following inequalities for the k-Riemann-Liouville fractional integrals: .

Conclusion
In this work, the authors established Hermite-Hadamard type inequalities for harmonically convex functions by using generalized fractional integrals. Furthermore, the authors proved some extensions of newly proved inequalities without using the condition of harmonic convexity for the functions. It is an interesting and new problem, and the upcoming researchers can offer similar inequalities for harmonically convex functions on the co-ordinates via generalized fractional integrals in their future research.