New Riemann–Liouville fractional Hermite–Hadamard type inequalities for harmonically convex functions

In this paper, we proved two new Riemann–Liouville fractional Hermite–Hadamard type inequalities for harmonically convex functions using the left and right fractional integrals independently. Also, we have two new Riemann–Liouville fractional trapezoidal type identities for differentiable functions. Using these identities, we obtained some new trapezoidal type inequalities for harmonically convex functions. Our results generalize the results given by İşcan (Hacet J Math Stat 46(6):935–942, 2014).


Introduction
Let f : I ⊆ R → R be a convex function defined on the interval I of real numbers and a, b ∈ I with a < b. The inequality , then f is harmonically convex on [a, b] if and only if g is convex on [ 1 b , 1 a ] (see [3]). In [4],İşcan gave Hermite-Hadamard type inequalities for harmonically convex functions as follows. Theorem 1.3 [4] Let f : I ⊂ R\ {0} → R be a harmonically convex function and a, b ∈ I with a < b. If f ∈ L [a, b], then the following inequalities hold: For some similar studies with this work about harmonically convex functions, readers can see [1][2][3][4][5][6][8][9][10]13,14] and references therein.
The following definitions of the left-and right-side Riemann-Liouville fractional integrals are well known in the literature.
In [6],İşcan and Wu presented Hermite-Hadamard type inequalities for harmonically convex functions in fractional integral form as follows.
We recall the following inequality and special function which are known as hypergeometric function: The following properties of convex functions are used in the forward results. Definition 1.6 [15, page 12] A function f defined on I has a support at x 0 ∈ I if there exists an affine functions for all x ∈ I . The graph of the support function A is called a line of support for f at x 0 . Theorem 1.7 [15, page 12] f : (a, b) → R is a convex function if and only if there is at least one line of support for f at each x 0 ∈ (a, b).
As much as we know, there are so many studies in the literature for Hermite-Hadamard type inequalities using the left and right fractional integrals together (such as Riemann-Liouville fractional integrals, Hadamard fractional integrals and conformable fractional integrals). In all of them, the left and right fractional integrals are used together. As much as we know, the studies [11,12] are the first two works using only the right fractional integrals or the left fractional integrals.
In this paper, our aim is to obtain new Riemann-Liouville fractional Hermite-Hadamard type inequalities using only the right or the left fractional integrals separately for harmonically convex functions. Also, we improve the fractional Hermite-Hadamard type inequalities for harmonically convex functions (1.4).
Hence, using Theorem 1.7, there is at least one line of support . From (2.2) and harmonically convexity of f , we have for all t ∈ [0, 1]. Multiplying all sides of (2.3) with αt α−1 and integrating over [0, 1] respect to t, we have With a combination of (2. where h (x) = 1 x and α > 0.
Proof Similar to the proof of Theorem 2.1, there is at least one line of support . From (2.8) and harmonically convexity of f , we have for all t ∈ [0, 1]. Multiplying all sides of (2.9) with αt α−1 and integrating over [0, 1] respect to t, similarly we have (2.7) and we omit the details.
, then the following inequality for the Riemann-Liouville fractional integrals holds: where h (x) = 1 x and α > 0.
Proof Adding the inequalities (2.1) and (2.7) side by side, then multiplying the resulting inequalities by 1 2 , we have the inequalities (2.10).

Corollary 2.7
The left-hand side of (2.10) is better than the left-hand side of (1.4).

Lemmas
In this section, we will prove two new identities used in the forward results.
Then the following equality for the right Riemann-Liouville fractional integral holds: Proof It can be proved directly by applying the partial integration to the right-hand side of Eq. (3.1) as follows: This completes the proof.
Proof Similar to the proof of Lemma 3.1, it can be proved directly by applying the partial integration to the right-hand side of Eq. (3.3) and we omit the details.
Proof Using Lemma 3.1, power mean inequality and harmonically convexity of | f | q , we have Calculating the appearing integrals in (4.2), we have a, b, α) .