Dynamical significance of generalized fractional integral inequalities via convexity

1 Department of Mathematics, University of Lahore, Sargodha Campus, Pakistan 2 Department of Mathematics, University of Sargodha, Sargodha, Pakistan 3 Department of Mathematics and Statistics, Hazara University, Mansehra, Pakistan 4 Department of Mathematics, College of Arts and Sciences, Wadi Aldawser, 11991, Prince Sattam bin Abdulaziz University, Saudi Arabia 5 Department of HEAS (Mathematics) , Rajasthan Technical University, Kota, India 6 King Khalid University, College of Science, Department of Mathematics, P. O. Box 9004, 61413 Abha, Saudi Arabia 7 Department of Mathematics, Faculty of Science, Al-Azhar University, 71524 Assiut, Egypt


Introduction
Fractional calculus is one of the renowned fields in recent research due to its inherent applications in various areas such as mathematical physics, fluid dynamics, mathematical biology etc. [1][2][3][4][5][6]. On the other hand, the fractional integral inequalities with the fractional operators are developed by many researchers because these inequalities are used to verify various results of applied problems [7,8]. In particular, the researchers [30][31][32][33] have recently studied many remarkable fractional integral inequalities and their applications. In [39], Mehmood et al. discussed the Hermite-Hadamard-Fejér inequality for fractional integrals involving preinvex functions. Mehreen and Anwar [40] estimated he Hermite-Hadamard and Hermite-Hadamard-Fejér type inequalities for p-convex functions by utilizing conformable fractional integrals. In [41], Almutairi and Adem Klçman discussed new integral inequalities of Hermite-Hadamard type involving s-convexity and studied their properties. Budak [42] establish Hermite-Hadamard-Fejér type inequalities for convex function involving fractional integrals with respect to another function. The Hermite-Hadamard inequality is defined can be found in [9] for convex function by Ψ : I → R, m, n ∈ I, m < n, m, n ∈ R, I ⊆ R and is playing a significant role in the field of inequalities and are widely used by the researchers [10]. Fejér type integral inequalities can be found in [27][28][29] by for all m, n ∈ I and λ ∈ [0, 1] and I ⊆ R is a convex function and η : Ψ(I)×Ψ(I) → R is a bifunction.
for ∈ C, where Γ being the gamma function.
Definition 1.7. [43] The integral representation of gamma function is defined as for, (t) > 0.
Definition 1.11. The extended generalized Bessel-Maitland function is defined for µ, ν, η, ρ, γ, c ∈ C, In the recent era of research, the field of fractional calculus has gained more recognition due to its wide range of applications in different sciences [44,45]. Such new developments in fractional calculus motivate the researchers to establish some new innovative ideas to unify the fractional operators and propose new inequalities involving new fractional operators.
This paper aims to obtain Hermite Hadamard and Fejér inequalities using generalized fractional integral having extended generalized Bessel-Maitland function as its kernel.
The structure of the paper follows: In section 2, we present Hermite-Hadamard inequalities for convex function using generalized fractional operator. Section 3 is devoted to Trapezoid type inequalities related to Hermite-Hadamard inequalities. Fejér type inequalities for (η 1 , η 2 )-convex function using the generalized fractional operator are presented in section 4.

Hermite-Hadamard inequalities
In this section, we obtain the Hermite-Hadamard inequalities for convex function using generalized fractional operator as follows: If Ψ is an increasing function on [u, v], then for the generalized fractional integrals defined in definition 1.12, we have Proof. By the convexity of Ψ on the interval [u, v], let x, y ∈ [u, v] with t = 1 2 , we have where if we takes Multiplying both sides by (1 − t) v J µ,ξ,m,σ,c v ,η,ρ,γ (ω(1 − t) µ ; p) and integrating the resulting inequality on [0, 1] with respect to t, we have By making suitable substitutions in inequality (2.1),we obtain For second part of inequality, again using the convexity of Ψ, Addition of these inequalities, gives

Multiplying both sides by
and integrating the resulting inequality on [0, 1] with respect to t, we get Making substitution in the integrals involved leads to

Trapezoid inequalities related to the Hermite-Hadamard inequalities
The Trapezoid type inequalities related to the Hermite-Hadamard inequalities are presented in this section.
. Then for the generalized fractional integrals defined in definition (1.12), we have Proof. If we consider the integral Let Firstly, we consider I 1 Integrating by parts, we have On the same lines, we get Multiplying by v−u 2 , we have the required result. By using Lemma 3.1, we present the following theorem. ). Also, suppose that |Ψ | is a convex function on I, then for the generalized fractional integrals in definition 1.12, we have where v ≥ 0.
Proof. If we consider the following integral expression Solving the integrals involved by using integrating by parts method, we obtain

Fejér type inequalities for
Here, we present Fejér type inequalities for (η 1 , η 2 )-convex function by using the generalized fractional operator in definition 1.12.
for all t 1 , t 2 ∈ [0, 1]. Then for the generalized fractional integrals defined in definition (1.12), the following Fejér type inequality holds: Proof. By the (η 1 , η 2 )-convexity of the function Ψ and using (4.1),we get Now by adapting the same procedure as above, we obtain Using the generalized fractional integral operators defined in 1.12, we have By using the inequalities (4.2) and (4.3), we proceed or Again by simplification and using the above mentioned substitution as well as the symmetry of Φ to u + 1 2 η 1 (v, u) leads to the following It follows that
which is Fejér inequality for generalized fractional integral can be obtained by considering the function Ψ to be η-convex.
which is Fejér inequality for generalized fractional integral can be obtained by considering the function Ψ to be preinvex convex.
which is Fejér type inequality for generalized fractional integral can be obtained by considering the function Ψ to be convex.

Mid point and trapezoid type inequalities related to Hermite-Hadamard type inequalities
In the section, we discuss the midpoint and trapezoid type inequalities connected to Hermite-Hadamard inequalities for the function whose absolute value of the derivative is (η 1 , η 2 )-convex function. The following lemma will help us in the next result.
Lemma 5.1. Let a function Ψ : I → R with I ∈ R , Ψ ∈ L 1 [u, u + η 1 (v, u)] be a differentiable function where I is taken to be an open invex set with respect to η 1 : I × I → R with η 1 (v, u) > 0, for u, v ∈ I. Then for the generalized fractional integrals defined in definition 1.12, we have Solving the integrals by using integrating by parts method leads to Similarly, Similarly, Adding I 1 , I 2 , I 3 and I 4 , we proceed to the desired result as, Next, we present mid-point type inequalities related to Hermite-Hadamard inequalities: for 0 < v + µn ≤ 1.

Concluding remarks
Various researchers have studied integral inequalities due to their wide applications in both pure and applied mathematics. This paper discussed the new version of integral inequalities such as Hermite-Hadamard type and trapezoid type inequalities for the convex function by utilizing generalized fractional integrals concerning the extended Wright generalized Bessel function as a kernel. Also, we established new mid-point type and trapezoidal type integral inequalities for (η 1 , η 2 )-convex function related to Hermite-Hadamard and Fejér type inequalities. All the inequalities presented in this paper are more general than the inequalities available in the literature, which can easily observe from the corollaries.