On Inequalities of Trapezium Type Via Fractional Integrals Operators

In this article, we get solutions of some integral inequalities of Hermite-Hadamard type and using the approach of ( ψ , h )-Convex function by the way of Riemann-Liouville Fractional integrals and Katugampola Fractional integral operators.


Introduction
The concept of convexity has played a dominant role and has gotten special attention by many researchers in various places of pure and applied sciences. It is noticed that the convex (concave) function is one of the most significant function which is furthermore generalized day by day see the References [5,7,8,9,13,15]. One of the significant generalization from these references is ψ-Convex function which is furthermore generalized by using the concept of Raina's function as (ψ,h)-Convex function in [13] by R. Saima. In literature, there are so many results related with convex or generalized convex function in inequalities, one of the popular inequality is Hermite-Hadamard inequality, which is widely seen in the mathematical literature. The concept of (ψ,h)-Convexity provides a powerful tools in proving a large scale of inequalities. Several generalizations and extensions concerning to the below inequality have been proved by many researchers see the Reference [2] Dragomir and Agarwal. Let g be a convex function on the finite interval [v 1 , v 2 ], then Different results have been established using this integral inequality by connecting it with Riemann-Liouville fractional integrals see for instance [4]. The above inequality has never ceased to fascinate researchers, several variants, extensions, generalization and improvements have been set up.
In [2] Dragomir and Agarwal derived the following Hermite-Hadamard type inequality in this form Email addresses: malik.muddassar@gmail.com (M. Muddassar), tahirajabeen14@gmail.com (T. Jabeen), hira.perveen96@gmail.com (H. Perveen) In [7] Kermausuor and Nwaeze proved the following inequality as by connecting it with Katugampola fractional integral for strongly η− quasi-convex function. Different researchers has worked on different integral inequalities such as Hermite and Hadamard type, Simpson and Fejer type inequalitites, using the approach of generalizations of convex function such as η−convex, Quasi-convex, (η, ψ) − convex and preinvex functions. For recent references see [1,4,5,6,7,9,10,13]. The integral inequalities having different fractional integral operators such as Riemann-Liouville, katugampola and k-integral operator which have been considered in [4,7,14] respectively. A Since work in this direction has received many more attention, we try to introduce some useful formulations in this article such as the known results involving different fractional integrals and fractional integral operators become more generalized and comprehensible and these results give some new ideas to the upcoming researchers. This section contains different basic definitions of convexity and operators,as well as some useful results that will be necessary for the development of the present work.
Definition 1.4. [7] The left and right-sided Riemann-liouville fractional integrals of order α > 0 of g are defined by where k is positive and Γ k is the k-Gamma function defined by Then the left and right-sided Katugampola fractional integrals of order α > 0 of g∈ X p c (v 1 , v 2 ) are defined by The left and right-sided Hadamard fractional integrals of order α > 0 of g are defined by Where I, J, H are the Katugampola, Riemann-liouville and Hadamard fractional integral operators respectively. Similarly, these results hold for right-sided operators. Theorem 1.8. For ρ = k = 1 .Then for α > 0 Where I and J are the Katugampola and Riemann-liouville fractional integral operators respectively. Theorem 1.9.
In particular, this article have some generalizations about the Hermite-Hadamard inequalities using the approach of (ψ,h)-Convex function and fractional integral operators.
Adding (2.1) and (2.2), multiply both sides by ζ α k −1 and then integrate from 0 to 1 we get Using Definition 1.4 we get Which is the required result.
Adding (2.6) and (2.7) ,multiply both sides by ζ αρ−1 and integrate from 0 to 1 we get , 2 ))dζ Using the Definition 1.4 we get which is the required result.
Remark 2.8. Using the assumption (v in Theorem 2.7, we will get Lemma 3 in [7].
then the following inequality holds if the fractional integrals exists: Proof. The prove follows directly by using the the definition of ψ-Convex function and then integrating.
If |g′| q be a ψ-Convex function for q > 1 then the following inequality holds: Where I is katugampola fractional integral operator and is Raina's function.
Proof. As Theorem 2.7 Using Holder's inequality and the ψ-Convexity of |g′| q , we get This completes the proof.
Proof. Using Holder's inequality and the ψ-Convexity of |g′| q in Theorem 2.7, we get This completes the proof.

Application to Special Means
[13] For some positive real numbers v 1 , v 2 (v 1 ̸ = v 2 ), v 2 ≥ v 1 , we shall assume the following special means.
1 and a function g : [v 1 , v 2 ] −→ R = x n , (n ∈ N) we will have the above inequality.

Conclusion
In this article, we consider a new generalized form of convex functions which is known as ψ-convex functions. In the achievement of our target, we have derived some new inequalities that deduced from the definition of different fractional integral operators and the use of ψ-Convex functions. The results for ψ-convex functions generalizing and enhancing the results and inequalities which are already existed in mathematical literature. Others antecedently got for the Riemann-liouville fractional integrals. Specifically, our results focusing on the most popular integral inequality which is known as Hermite-Hadamard inequality and three famous fractional integrals. These results will serve as a motivation and prove benefical for future work in this field. For the suitable selection of the function h(ζ ) one can discover so many numerous results as particular cases. This shows that the idea of generalized convexity is extremely wide and unifying. It is expected that this article will provide new directions and ideas in fractional operators, special functions and related fields.