Fractional Integral Inequalities for Exponentially Nonconvex Functions and Their Applications

: In this paper, the authors deﬁne a new generic class of functions involving a certain modiﬁed Fox–Wright function. A useful identity using fractional integrals and this modiﬁed Fox– Wright function with two parameters is also found. Applying this as an auxiliary result, we establish some Hermite–Hadamard-type integral inequalities by using the above-mentioned class of functions. Some special cases are derived with relevant details. Moreover, in order to show the efﬁciency of our main results, an application for error estimation is obtained as well.

Integral inequalities play a major role in the fields of differential equations and applied mathematics (see [10,11]). Moreover, they are linked with such other areas as differential The following class of functions was introduced by Awan et al. (see [20]) and was demonstrated to play an important role in optimization theory and mathematical economics.
The most significant inequality about a convex function ψ on the closed interval [ξ 1 , ξ 2 ] is the Hermite-Hadamard integral inequality (that is, the trapezium inequality). This two-sided inequality is expressed as follows: The two-sided inequality (1) has become a very important foundation within the field of mathematical analysis and optimization. Several applications of inequalities of this type have been derived in a number of different settings (see [21][22][23][24][25][26][27][28][29]).
In the context of fractional calculus, the standard left and right-sided Riemann-Liouville (RL) fractional integrals of order α > 0 are given, respectively, by where ψ is a function defined on the closed interval [ξ 1 , ξ 2 ] and Γ(·) is the classical (Euler's) gamma function.
There are many directions in which one can introduce a new definition of fractional derivatives and fractional integrals, which are related to or inspired by (for example) the RL definitions (see [35,36]), with reference to some general classes into which such fractional calculus operators can be classified. In applied mathematics, it is important to consider particular types of fractional calculus operators which are suited to the fractional-order modeling of a given real-world problem.
The following modified version of the Fox-Wright function p Ψ q (z) in (3) was introduced, as long ago as 1940, by Wright [38] (p. 424), who partially and formally replaced the Γ-quotient in (3) by a sequence {σ(n)} ∞ n=0 based upon a suitably-restricted function σ(τ) as follows (see also [39], where the same definition is reproduced without giving credit to Wright [38]): If, in Wright's definition (6) from 1940 (see [38] (p. 424)), we take ρ = ς = 1 and then Wright's definition (6) would immediately yield the familiar Fox-Wright hypergeometric function p Ψ q (z) defined by (3). The one-and two-parameter Mittag-Leffler functions E α (z) and E α,β (z), and indeed also almost all of the parametric generalizations of the Mittag-Leffler type functions, can be deduced as obvious special cases of the Fox-Wright hypergeometric function p Ψ q (z) defined by (3) (see [40] for details).
We are now in the position to introduce a new generic class of functions involving the modified Fox-Wright function F σ ρ,ς (·).
Our paper has the following structure: in Section 2, we first find a useful identity using fractional integrals with two parameters λ and µ involving the modified Fox-Wright function F σ ρ,ς (·). Applying this as an auxiliary result, we give some Hermite-Hadamardtype integral inequalities pertaining to exponentially ( 1 , 2 , h 1 , h 2 )-nonconvex functions, and some special cases are derived in details. In Section 3, the efficiency of our main results is demonstrated with an application for error estimation. Section 4 presents the conclusion of this paper.

Main Results and Their Consequences
The following notations are used below: is the interior of the closed interval ∆ with 1 , 2 ∈ R. We denote by L 1 (∆) the space of integrable functions over ∆. We need to prove the following basic lemma.
Proof. We define where which, upon integrating by parts, would yield Similarly, we find that Substituting from (9) and (10) into (8), we obtain the desired result (7).
From Lemma 1, we can derive the following case: Our first main result is stated as Theorem 1 below.
Some corollaries and consequences of Theorem 1 are listed below: Corollary 1. Upon setting α = 1, Theorem 1 yields

Corollary 4.
Taking in Theorem 1, the following inequality is deduced: where
Our second main result is stated as Theorem 2 below.
We now state several corollaries and consequences of Theorem 2.

Corollary 8.
Choosing Theorem 2 is reduced to the following inequality: where Theorem 2 yields the following inequality: where

Remark 5.
If we choose 1 = 2 = 0 in our results in this paper, then all of the consequent results will hold true for the (h 1 , h 2 )-nonconvex functions.

Application
In this section, we present an application involving a new error estimation for the trapezoidal formula by using the inequalities obtained in Section 2. We fix the parameters ρ and ς. We also suppose that the bounded sequence {σ( )} ∞ =0 of positive real numbers is given.
For λ, µ ∈ (0, 1], let us define where R(U, ψ) is the remainder term and From the above notations, we can obtain some new bounds regarding error estimation.

Remark 6.
In view of Remark 2, we can establish new error estimations by using Proposition 1 and Proposition 2.

Conclusions
In this paper, the authors have defined a new generic class of functions involving the modified Fox-Wright function F σ ρ,ς (·) as well as the so-called exponentially ( 1 , 2 , h 1 , h 2 )nonconvex function. A useful identity has also been found by using fractional integrals and the function F σ ρ,ς (·) with two parameters λ and µ. We have established some Hermite-Hadamard-type integral inequalities by using the above class of functions and the aforementioned identity as an auxiliary result. Several special cases have been deduced as corollaries including relevant details. We have also outlined the derivations of several other corollaries and consequences for the interested reader. The efficiency of our main results has been shown by proving an application for error estimation.