Hermite-Hadamard-Fej´er type fractional inequalities relating to a convex harmonic function and a positive symmetric increasing function

: The purpose of this article is to discuss some midpoint type HHF fractional integral inequalities and related results for a class of fractional operators (weighted fractional operators) that refer to harmonic convex functions with respect to an increasing function that contains a positive weighted symmetric function with respect to the harmonic mean of the endpoints of the interval. It can be concluded from all derived inequalities that our study generalizes a large number of well-known inequalities involving both classical and Riemann-Liouville fractional integral inequalities.


Introduction
We recall that an interval χ ⊂ R is convex if for all x, y ∈ χ, we have tx + (1 − t) y ∈ χ, where t ∈ [0, 1] and a function f : χ → R is convex if for all x, y ∈ χ, the inequality holds. A function f : χ → R is concave if the inequality (1.1) holds in opposite direction. Convexity is essential to understanding and solving problems pertaining to fractional integral inequalities because of its properties and definition, and it has recently gained in importance. Convex functions have yielded several new integral inequalities, as evidenced by [2,3,8,11,14,15,21,24,43,[45][46][47][48]. Hermite-integral Hadamard's inequalities are most commonly encountered when searching for comprehensive inequalities: where the function f : χ → R is convex on χ and f ∈ L 1 ([u, v]).
There are the following two classical fractional integral inequalities which are defined by Hermite-Hadamard type inequalities: where the function f : χ → R is positive, convex on χ and f ∈ L 1 ([u, v]). The left-sided and right-sided Riemann-Liouville fractional integrals J ν u + f (v) and J ν v − f (u) of order ν > 0 in (1.3), are defined respectively as (see [5,22]): The extended inequalities for (1.2) and (1.5) are fractional integral inequalities of the Fejér and Hermite-Hadamard-Fejér types, and the results are as follows: (1.5) respectively, where f is as defined above, the function ζ : [u, v] → [0, ∞) is integrable symmetric with respect to u+v 2 , that is and Γ (ν) is the Gamma function defined as Γ (ν) = ∞ 0 x ν−1 e −x dx, Re (ν) > 0. In addition to being able to be generalized, convexity and convex functions have several generalizations. One of those generalizations is the concept of harmonic convexity and harmonic convex functions, which can be defined as follows: Using harmonic-convexity, the Hermite-Hadamard type yields the following result. Theorem 1.2. Let f : χ ⊆ R\ {0} → R be a harmonically convex function and u, v ∈ χ with u < v. If f ∈ L ([u, v]) then the following inequalities hold: Harmonic symmetricity of a function is given in the definition below.
Fejér type inequalities using harmonic convexity and the notion of harmonic symmetricity were presented in Chan and Wu [7].
Hermite-Hadamard's inequalities for harmonically convex functions in fractional integral form were proved in [19].
If f is a harmonically convex function on [u, v], then the following inequalities for fractional integrals hold: with ν > 0 and g (x) = 1 x . Hermite-Hadamard-Fejér inequality for harmonically convex function in fractional integral form were obtained byİşcan et al. in [18].
If ζ : [u, v] → R is nonnegative, integrable and harmonically symmetric with respect to 2uv u+v , then the following inequalities for fractional integrals holds: with ν > 0 and g (x) = 1 Left-sided and right-sided Riemann-Liouville fractional integrals are generalized in the definition given below: Definition 1.7. [20] Let τ (x) be an increasing positive and monotonic function on the interval (u, v] with a continuous derivative τ (x) on the interval (u, v) with τ (x) = 0, 0 ∈ [u, v]. Then, the left-side and right-side of the weighted fractional integrals of a function f with respect to another function τ (x) on [u, v] of order ν > 0 are defined by: The following observations are obvious from the above definition: • If τ (x) = x and ζ (x) = 1, then the weighted fractional integral operators in the Definition 1.7 reduce to the classical Riemann-Liouville fractional integral operators. • If ζ (x) = 1, we get the fractional integral operators of a function f with respect to another function τ (x) of order ν > 0, defined in [1,38] as follows: The study analyzes several inequalities of the Hermite-Hadamard-Fejér type through weighted fractional operators with positive symmetric weight function in the kernel.

Main results
Throughout this paper, R denotes the set of all real numbers, χ ⊂ R denotes an interval. In addition. for u, v ∈ J • with u < v, the functions θ u,v , θ v,u : [0, 1] → R are defined as: We start this section with the following Lemma which will be used repeatedly in the sequel.
Lemma 2.1. The following results hold: be an integrable function and symmetric with respect to 2uv u+v , then we have ζ θ u,v (t) = ζ θ v,u (t) (2.1) be an integrable and symmetric function with respect to 2uv u+v , then we have for ν > 0 for each t ∈ [0, 1] and then Hence by the definition of harmonic symmetry, we obtain (ii) By using the harmonic symmetric property of ζ, we have for all t ∈ τ −1 1 v , τ −1 1 u . From this and by setting 1 Theorem 2.2. Let f : I ⊆ (0, ∞) → R be an L 1 convex function with 0 < u < v, u, v ∈ I and ζ : [u, v] → R be an integrable, positive and weighted symmetric function with respect to 2uv u+v . If τ is an increasing and positive function on [u, v) and τ (x) is continuous on (u, v), then, we have for ν > 0: Choosing x = ζ θ u,v (t) and y = ζ θ v,u (t) , we obtain Multiplying both the sides by t ν−1 ζ θ u,v (t) and then integrating the resultant with respect to "t" over [0, 1], we obtain To prove the first inequality in (2.3), we need to use (2.2) By evaluating the weighted fractional operator, one can observe that Using (2.5) and (2.6) in (2.4), we obtain Thus the first inequality is proved.
To prove the second inequality we use the convexity of f Multiplying both the sides by t ν−1 ζ θ u,v (t) and then integrating the resultant with respect to "t" over [0, 1], we obtain Then by using (2.1) where ζ J ν u+ and ζ J ν v− are the left and right weighted Riemann-Liouville fractional operators of order ν > 0, defined by (2) τ (x) = x and ν = 1, then inequality (2.3) transforms into inequality (1.8).   (u, v), then, we have for ν > 0: By integration by parts, making use of Lemma 2.1, and definitions (1.11) and (1.12), we obtain and Thus, we obtain from (2.13) that Which is the required result.

Remark 2. Particularly, in Lemma 2.3, if we take:
(1) τ (x) = x, then equality (2.11) becomes where ζ J ν u+ and ζ J ν v− are defined in Remark 1. (2) τ (x) = x and ζ(x) = 1, then equality (2.11) becomes We will use the following notations for the rest of this section: Proof. According to (2.11) of Lemma 2.3, we obtain We know that ζ is a harmonic symmetric with respect to 2uv u+v , we observed that Let us evaluate the first integral in (2.25) (2.26) and the value of the second integral in (2.25) is given below (2) τ (x) = x and ζ(x) = 1, we get where u (ν, u, v) and v (ν, u, v) are defined as above.
Proof. Applying power-mean inequality to ( 2.20) and then using (2.22)-(2.24), we get Since for q ≥ 1 and t ∈ τ −1 1 Thus, now we are able to evaluate the second integral in (2.32) Remark 4. In Theorem 2.5, especially when we take u (ν, u, v) and v (ν, u, v) are defined in (1) of Remark 3.

Conclusions
In this study, we proved very important and interesting inequalities of Fejér type for a very fascinating generalized class of functions, namely, harmonic convex functions by using general weighted fractional integral operator which depends upon an increasing function. The results of our study not only generalize a number of findings obtained in [17][18][19] but one can obtain a number of new results by choosing a increasing function involved. The results can also be an inspiration for young researchers as well as researcher already working in the field of fractional integral inequalities and can further open up new directions of research in mathematical sciences.