Estimations of Upper Bounds for n-th Order Differentiable Functions Involving χ-Riemann–Liouville Integrals via c-Preinvex Functions

In recent years, several new generalizations of the classical concepts of convexity have been proposed in the literature. For example, Hanson [1] introduced the notion of differentiable invex functions, without calling them as invex, in connection with their special global optimum behaviour. It was Craven [2] who introduced the term invex for calling this class of functions, due to their property described as invariance by convexity. &e concept of invex sets is defined as follows. A set Cμ ⊂ R is said to be invex with respect to bifunction μ(·, ·): Cμ × Cμ↦R, if


Introduction and Preliminaries
A set C ⊂ R is said to be convex, if (1) A function 5: C ↦ R is said to be convex, if In recent years, several new generalizations of the classical concepts of convexity have been proposed in the literature. For example, Hanson [1] introduced the notion of differentiable invex functions, without calling them as invex, in connection with their special global optimum behaviour. It was Craven [2] who introduced the term invex for calling this class of functions, due to their property described as invariance by convexity. e concept of invex sets is defined as follows.
Weir and Mond [4] introd3uced the concept of preinvex functions as follows.
A function 5: C μ ↦ R is said to be preinvex with respect to bi-function μ(·, ·): Note that if we take μ(♭ 2 , ♭ 1 ) � ♭ 2 − ♭ 1 , then from the class of preinvex functions, we recapture the class of classical convex functions.

(5)
It has been observed that the class of c-preinvex functions generalizes several other classes of preinvexity and convexity. For example: (1) If we take c(v) � 1, then we have the class of classical preinvex function. (2) If we take c(v) � v − 1 , then we have the definition of P-preinvex function (see [6]). (3) If we take c(v) � v s− 1 where s ∈ (0, 1), then we have the class of s-preinvex functions of Breckner type (see [6]). (4) If we take c(v) � v − s− 1 , then we have the class of s-Godunova-Levin-Dragomir type of preinvex functions (see [9]). (5) If we take c(v) � 1 − v, then we have the definition of tgs-preinvex functions (see [8]).
It is obvious that if we take μ(♭ 2 , ♭ 1 ) � ♭ 2 − ♭ 1 in the above discussed special cases, then we can recapture the classes of classical convexity. eory of convexity has played significant role in the development of theory of inequalities. Many famously known results in theory of inequalities can easily be obtained using the convexity property of the functions. e Hermite-Hadamard inequality which provides us a necessary and sufficient condition for a function to be convex is one of the most studied results pertaining to convexity. It reads as follows.
Let 5: C � [♭ 1 , ♭ 2 ]↦R be a convex function; then, In recent years, several new extensions and generalizations for the Hermite-Hadamard inequality have been obtained in the literature. For example, Noor [10] obtained a new refinement of the Hermite-Hadamard inequality using the class of preinvex functions. Awan et al. [8] obtained its new version by utilizing the class of c-preinvex functions. e authors have also discussed various applications for some special means. Sarikaya et al. [11] introduced a new dimension by introducing the fractional analogue of the Hermite-Hadamard inequality. e idea of Sarikaya and his co-authors has attracted many inequalities experts and consequently a variety of new fractional analogues of classical inequalities have been obtained in the literature using different variants of classical concepts of fractional calculus and also by different generalizations of classical convexity. For example, Hwang et al. [12] obtained different refinements and extensions of the Hermite-Hadamard inequality via fractional integrals. Turhan et al. [13] obtained Hermite-Hadamard type of inequalities via n-times differentiable convex functions involving Riemann-Liouville fractional integrals. Wu et al. [14] obtained fractional analogues of k-th order differentiable functions involving Riemann-Liouville integrals via higher order strongly h-preinvex functions. Zhang et al. [15] obtained new k-fractional integral inequalities containing multiple parameters via generalized (s, m)-preinvexity. e aim of this paper is to derive a new integral identity involving n-times differentiable functions and χ-Riemann-Liouville fractional integrals. Some associated estimates of upper bounds involving c-preinvex functions are also obtained. In order to relate some unrelated results, several special cases are discussed. is shows that our results are more generalized and quite unifying. In order to show the significance of our obtained results, we also present applications to special means of real numbers. We hope that the ideas of this paper will inspire interested readers working in this field.
Before we proceed further, let us recall some basic preliminaries from fractional calculus. ese preliminaries will be helpful during the study of this paper. Definition 1. (see [16]).
respectively, and Γ(α) is the gamma function. Also, we define J 0 Mobeen and Habibullah extended the notion of Riemann-Liouville fractional integrals and introduced the concept of χ-Riemann-Liouville fractional integrals.

Auxiliary Result
We now derive a key lemma which will be helpful in obtaining the coming results of the paper.

Lemma 1.
Let n ≥ 1 and 5: where Proof. We prove this result by using the mathematical induction principle. e case for n � 1 is obvious. Suppose Lemma 1 holds for n − 1, that is, Now, we prove (11) for n. Integrating by parts, we have Mathematical Problems in Engineering 3 is completes the proof. (i) If we take n � 1 in (11), then we have the following identity: (ii) If we take n � 2 to (11), then On the other hand, one can easily see that One can easily see that (16) and (17) are identical. (iii) If we take χ � 1 to (11), then we have where

Main Results
We now derive our main results of the paper.

Mathematical Problems in Engineering
Proof. Using Lemma 1 and the c-preinvexity of |5 (n) |, we have n is even, Mathematical Problems in Engineering is completes the proof.

□
We now discuss some special cases of eorem 1.
(i) If c(v) � 1, then eorem 1 reduces to the following result for the class of preinvex function.

Corollary 1. Under the assumptions of eorem
Mathematical Problems in Engineering

Mathematical Problems in Engineering
Proof. Using Lemma 1, Hölder's integral inequality, and c-preinvexity of |5 (n) | q , we have n is even,

Mathematical Problems in Engineering 11
is completes the proof.

□
We now discuss some special cases of eorem 2.
(i) If c(v) � 1, then eorem 2 reduces to the following result for the class of classical preinvex function.

Mathematical Problems in Engineering 13
(v) If c(v) � 1 − v, then eorem 2 reduces to the following result for the class of tgs-preinvex function.
□ Now we will discuss some special cases of eorem 3.
(i) If c(v) � 1, then eorem 3 reduces to the following result for the class of classical preinvex function.

Conclusion
We have derived a new fractional integral identity by using the χ-fractional integral and n-times differentiable functions. Using this identity as an auxiliary result, we have obtained some new associated upper bounds involving c-preinvex functions. Several new special cases are also discussed in detail which show that our results represent significant generalizations and are quite unifying. In order to show the significance of our theocratical results, we also presented applications to special means of our obtained results.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.