Weighted Midpoint Hermite-Hadamard-Fejér Type Inequalities in Fractional Calculus for Harmonically Convex Functions

: In this paper, we establish a new version of Hermite-Hadamard-Fejér type inequality for harmonically convex functions in the form of weighted fractional integral. Secondly, an integral identity and some weighted midpoint fractional Hermite-Hadamard-Fejér type integral inequalities for harmonically convex functions by involving a positive weighted symmetric functions have been obtained. As shown, all of the resulting inequalities generalize several well-known inequalities, including classical and Riemann–Liouville fractional integral inequalities.


Introduction
The theory of convex functions is an essential tool in various fields of pure and applied sciences. There is also a close connection between the theory of convex functions, the theory of inequalities, and fractional differential equations. At the same time, fractional differential equations are one of the most studied fields of mathematics due to their application in the real world. Many inequalities are proved for convex functions but, the most known from the related literature is Hermite-Hadamard inequality.
A function F : [θ 1 , θ 2 ] ⊂ R → R on an interval of real line is said to be convex, if for all θ 1 , θ 2 ∈ I and τ ∈ [0, 1], then The Hermite-Hadamard integral inequality is a well-known inequality in the subject of convex functional analysis. It has an interesting geometric representation with numerous important applications. The extraordinary inequality states that if F : I → R is a convex mapping on the interval I of real numbers and θ 1 , θ 2 ∈ I with θ 1 < θ 2 , then Inequality (2) was introduced by C. Hermite [1] and investigated by J. Hadamard [2] in 1893. Both inequalities hold in the inverted direction if F is concave. Many mathematicians have paid considerable attention to Hermite-Hadamard inequality due to its quality and integrity in mathematical inequality. For significant developments, modifications, and consequences regarding the Hermite-Hadamard uniqueness property and general convex function definitions, for essential details, the interested reader would like to refer to [3][4][5][6][7] and references therein. Fractional calculus and applications have application areas in many different fields such as physics, chemistry and engineering, and mathematics. Applying arithmetic in classical analysis in the fractional analysis is very important in obtaining more realistic results in solving many problems. Many real dynamical systems are better characterized using non-integer order dynamic models based on fractional computation. While integer orders are a model that is not suitable for nature in classical analysis, fractional computation in which arbitrary orders are examined enables us to obtain more realistic approaches. Regarding some papers dealing with fractional integral inequalities via different types of fractional integral operators, we refer readers to [8][9][10][11][12][13][14][15][16].
The midpoint Hermite-Hadamard inequality was discovered by Sarikaya and Yildirim [19] after expending the essential area of the integral inequalities in (2) and (3) where the function F : [θ 1 , θ 2 ] → R is convex and continuous. In [20], I. G. Macdonald gave the following definition.
and it is symmetric with respect to Fejér proposed the following generalization of Hadamard inequality in 1906 (see [21]): → R be a positive, integrable and symmetric to θ 1 +θ 2 2 . Then the following inequality holds: The inequality (6) is well-known as the Fejér-Hadamard inequality in the literature. In the concept of Riemann-Liouville fractional integrals,İ.İşcan [22] discovered the endpoint version of (6), which is also the extension of (3). As a result, the final inequalities are shown as follows: where F is convex and continuous, G is symmetric and belongs to L 1 [θ 1 , θ 2 ], (see Definition 1).
In [23],İ.İşcan gave definition of harmonically convex functions and established following Hermite-Hadamard type inequality for harmonically convex functions as follows: Definition 2. Let I ⊂ R\{0} be an interval of nonzero real numbers. Then a function F : I → R is said to be harmonically convex if holds for all θ 1 , θ 2 ∈ I and τ ∈ [0, 1].
In [24], Latif et al. gave the following definition.
We recall the following special functions which are known as Beta and hypergeometric function respectively, (see [18]).
The polygamma function of order m is a meromorphic function on the complex numbers C defined as the (m + 1)th derivative of the logarithm of the gamma function: Thus holds where ψ(ζ) is the digamma function and Γ(ζ) is the gamma function. When m > 0 and Re > 0, the integral representation of polygamma is given by The generalized hypergeometric function is given by a hypergeometric series, i.e., a series for which the ratio of successive terms can be written as c k+1 c k = (k + θ 1 )(k + θ 2 )...(k + θ p ) (k + π 1 )(k + π 2 )...(k + π q )(k + 1) .

Auxiliary Results
Lemma 1. If F : [θ 1 , θ 2 ] → R is integrable and harmonically symmetric with respect to (ii) For v > 0, we have and Then, using the hypotheses and Definition 3, we can obtain (14).
(ii) w has symmetry characteristic, then Hence, from above and setting 1 σ(τ) = , it follows that which brings the needed equality (15).
→ R be an L 1 harmonically convex function and w : [θ 1 , θ 2 ] → R is nonnegative, an integrable, and symmetric weighted function with respect to If σ is is an increasing and positive function from [θ 1 , θ 2 ) onto itself such that its derivative Multiplying both sides of (17) by κ v−1 w and integrating the resulting inequality with respect to κ over [0, 1], we obtain 2F 2θ 1 θ 2 From the left-hand side of the inequality in (18), we use (15) to obtain 1 2 It follows that We can demonstrate that by calculating the weighted fractional operators, Setting , one can deduce that As a consequence, When we use (19) and (20) in (18), we obtain the following result As a result, left inequality of (18) has been proven. The second inequality of (18) can be proved using the harmonically convex function of F . F 2θ 1 θ 2 Multiplying both sides of (22) by κ v−1 w 2θ 1 θ 2 κθ 1 +(2−κ)θ 2 and we obtain by integrating the resulting inequality in terms of κ on [0, 1].

Main Results
We can conclude the following Hermite-Hadamard-Fejér inequalities with the help of Lemma 2.
Proof. Using the Lemma 2 as well as properties of power mean inequality and the harmonically convex function of |F | q , we get As a result, we get where it is obvious that We can obtain the following integral after inserting (37) into (36) We can obtain the necessary result (33) by doing basic integral calculations based on inequality (38).

Theorem 9.
Suppose that all the conditions of Lemma 2 and |F | q is harmonically convex on [θ 1 , θ 2 ] with q > 1 and σ is an increasing and positive function from [θ 1 , θ 2 ) onto itself such that its derivative σ (τ) is continuous on (θ 1 , θ 2 ), for v > 0, then we have As a result, we get where it is obvious We obtain the following integral after inserting (43) into (42): We can obtain the necessary result (39) by doing basic integral calculations based on inequality (44).

Conclusions
In this paper, inequalities of the Hermite-Hadamard-Fejér type for harmonically convex functions in fractional integral forms are given in this study. Using weighted fractional integrals with positive weighted symmetric function kernels, an integral identity and various midpoint fractional Hermite-Hadamard-Fejér type integral inequalities for harmonically convex functions are also found.