Generalized fractional integral inequalities of Hermite–Hadamard type for harmonically convex functions

In this paper, we establish inequalities of Hermite–Hadamard type for harmonically convex functions using a generalized fractional integral. The results of our paper are an extension of previously obtained results (İşcan in Hacet. J. Math. Stat. 43(6):935–942, 2014 and İşcan and Wu in Appl. Math. Comput. 238:237–244, 2014). We also discuss some special cases for our main results and obtain new inequalities of Hermite–Hadamard type.


Introduction
The Hermite-Hadamard inequality introduced by Hermite and Hadamard, see [4], and [17, p. 137], is one of the best-established inequalities in the theory of convex analysis with a nice geometrical interpretation and several applications. These inequalities state the following.
If f : I → R is a convex function on the interval I of real numbers and a, b ∈ I with a < b, (1.1) Definition 1 ([8]) A function f : I ⊆ R\{0} → R is said to be a harmonically convex function if the following inequality holds: for all a, b in I and t in [0, 1]. If the inequality (1.2) holds in the reversed direction then f is called harmonically concave function.
İşcan [8] established the following identity and integral inequalities of Hermite-Hadamard type for harmonically convex functions. (1.3)

Lemma 1 ([8])
Let f : I ⊆ R\{0} → R be differentiable on I • (interior of I) and a, b ∈ I with a < b. If f ∈ L([a, b]), then the following identity holds: Now we recall some special functions and an inequality that will be needed in the sequel to establish our main results in this paper.
(a) The Beta function is defined as follows: The hypergeometric function is given as İşcan [10] also established the following identity and inequalities of Hermite-Hadamard type for harmonically convex functions via Riemann-Liouville fractional integrals.
, where a, b ∈ I with a < b. Then the following identity holds for the fractional integrals: where a -(f • g) and g is as given in Theorem 4.
for some fixed q ≥ 1, then we have the following inequality for the fractional integrals: and 0 < α ≤ 1.
If |f | q is harmonically convex function on [a, b] for some fixed q > 1, then we have the following inequality for the fractional integrals: for some fixed q > 1, then we have the following inequality for the fractional integrals: for some fixed q > 1, then we have the following inequality for the fractional integrals: For some similar studies with this work of harmonically convex functions, see ( [2,3,[13][14][15]18]). Now we recall the definition of left-and right-sided generalized fractional integrals given by Sarikaya and Ertuğral in [19] as follows: For details of the generalized fractional integrals see [19].
Some of the special cases of these generalized fractional operators are given as follows.
The main aim of this paper is to establish inequalities of Hermite-Hadamard type for harmonically convex functions using generalized fractional integrals. Some applications of the results presented in this paper are also obtained.

Main results
For brevity, throughout in this paper the following notations are used: We start with the following result.

Theorem 10 Let f : I ⊆ (0, +∞) → R be a function such that f ∈ L([a, b]). If f is harmonically convex function on [a, b], then the following inequalities hold for the generalized fractional integrals:
Proof Since f is harmonically convex function on [a, b], we have the following inequality: By changing the variables x = ab tb+(1-t)a and y = ab ta+(1-t)b , the inequality (2.4) becomes on both sides and integrating the resulting inequality with respect to t over [0, 1], we have ab t) t f ab tb + (1t)a dt which is first inequality of our desired result (2.3).
To prove the second inequality of (2. 3), note that f is harmonically convex function and hence the following inequalities hold for t ∈ [0, 1]: By adding (2.6) and (2.7), we have On multiplying the both sides of (2.8) by and integrating the result with respect to t on [0, 1], we obtain by changing the variables x = ab tb+(1-t)a and y = ab ta+(1-t)b , the inequality (2.9) becomes 1 a + I ϕ (f • g) Hence we have the proof of Theorem 10. (2.10)

Corollary 2
Under the assumptions of Theorem 10, if we take ϕ(t) = t(bt) α-1 , then we obtain the following inequalities:
Lemma 4 Let f : I ⊆ (0, +∞) → R be a differentiable function on I • such that f ∈ L ([a, b]), where a, b ∈ I with a < b. Then the following identity holds for the generalized fractional integrals: Proof Denote where (2.15) and Integrating (2.15) by parts, we have Similarly, using (2.16), we get Remark 5 Under the assumptions of Lemma 4, taking ϕ(t) = t α Γ (α) , the identity (2.13) reduces to identity (1.8).
Theorem 11 Let f : I ⊆ (0, +∞) → R be a differentiable function on I • such that f ∈ L ([a, b]), where a, b ∈ I with a < b. If |f | q is harmonically convex on [a, b] for some q ≥ 1, then the following inequality holds for the generalized fractional integrals: Proof From Lemma 4 and the well-known power mean inequality, we have which is our required inequality (2.19).

Corollary 4
Under the assumptions of Theorem 11, taking ϕ(t) = t α k kΓ k (α) , the inequality is obtained, where  and for harmonically convex function f (x) = x p+2 , where x > 0 and p ∈ (-1, +∞)\{0}; x 2 ln x, where x > 0, we obtain some new interesting inequalities using the special means. The details are left to the reader.

Conclusion
In this paper, we established inequalities of Hermite-Hadamard type for harmonically convex functions using generalized fractional integrals. Some special cases are provided as well. Finally, some application to special means are given. The results of the present paper can be applied in convex analysis, optimization and also different areas of pure and applied sciences.