Abstract

The main objective of this paper is to introduce a new class of preinvex functions which is called as -polynomial preinvex functions of a higher order. As applications of this class of functions, we discuss several new variants of trapezium-like inequalities. In order to obtain the main results of the paper, we use the concepts and techniques of -fractional calculus. We also discuss some special cases of the obtained results which show that the main results of the paper are quite unifying one.

1. Introduction

Convexity is one of the most important and natural notions in mathematics; it plays a significant role in various branches of pure and applied sciences [1–15]. For example, the set of feasible points in optimization theory is convex; the loss function used to measure the quality of solution in statistics is convex. In particular, many remarkable inequalities have been established via the convexity theory [16–32].

Recently, the classical concept of convexity has been extended and generalized in different directions. For instance, Hanson [33] introduced the notion of differentiable invex function but did not name it as invex; Craven [34] gave the term invex for this class of functions due to their property described as invariance by convexity. Mititelu [35] introduced the notion of invex set as follows.

Definition 1 (see [35]). Let be a nonempty set and be a real-valued function. Then is said to be invex with respect to if for all and .
We clearly see that the invexity reduces to the classical convexity if . Therefore, every convex set is an invex with respect to , but its converse is not true in general [35].
Weir and Mond [36] introduced the class of preinvex functions by use of the invex set.

Definition 2 (see [36]). Let be a nonempty invex set with respect to . Then the function is said to be preinvex with respect to if the inequality holds for all and .
Note that the preinvex function becomes the classical convex function if ; for more details regarding recent study on preinvexity property, we recommend the literature [37].
Very recently, Toplu et al. [38] introduced and investigated a new class of convexity named -polynomial convexity, and Karamardian [39] and Polyak [40] independently introduced the class of strongly convex functions. Strong convexity is the strengthening of convexity.

Definition 3 (see [39, 40]). A function is said to be strongly convex with respect to modulus if the inequality holds for all and .
Lin et al. [41] introduced higher order strongly convex functions to simplify the study of mathematical programs with equilibrium constraints.

Definition 4 (see [41]). Let . Then, the function is said to be -order strongly convex with respect to modulus if for all and .
If , then Definition 4 becomes Definition 3. Therefore, higher order strong convexity is a generalization of strong convexity. Lin et al. [41] proved that the higher order strong convexity of a function is equivalent to higher order strong monotonicity of the gradient map of the function.
It is well known that the Hermite-Hadamard inequality [42–45] is one of the most important and classical inequalities in convex function theory, which can be stated as follows.

Theorem 5. Let be a convex function. Then In recent decades, the fractional calculus has become a powerful tool in numerous branches of mathematics, physics, and engineering. The history of fractional calculus dates back to 1695 with the work of mathematicians such as L’Hospital and Leibniz, but the logical definitions were proposed by Liouville in 1834, Riemann in 1847, and Grünwald in 1867. Fractional calculus can be considered a super set of integer-order calculus, which has the potential to accomplish what integer-order calculus cannot. The classical form of the fractional calculus is given by the Riemann-Liouville integrals as follows.

Definition 6 (see [46]). Let , , and . Then, the -order Riemann-Liouville integrals and are defined by respectively, where is the Euler gamma function.
Sarikaya et al. [47] used the fractional calculus to obtain new variants of the Hermite-Hadamard inequality, which opened a new research area for the people who are working on the field of mathematical inequalities, particularly working on the inequalities involving convexity and its generalizations. For some more recent research work done in this direction, see [48–50].
Diaz et al. [51] introduced the generalized -gamma function and -beta function as follows. Making use of the generalized -gamma function, Sarikaya et al. [52] introduced the -Riemann-Liouville fractional integrals and discussed their properties and applications.

Definition 7 (see [52]). Let , , and . Then, the -order -Riemann-Liouville integrals and are defined by In recent years, several authors have used the concepts of -fractional calculus in obtaining new variants of fractional analogues of classical inequalities. For example, Huang et al. [53] obtained -fractional conformable analogues of Hermite-Hadamard’s inequality. Rahman et al. [54] obtained fractional analogues of the Gruss type inequalities using -conformable fractional integral operators.
The aim of the article is to obtain some new -analogues of trapezium-like inequalities involving a new class of functions called strongly -polynomial preinvex function of higher order. We also discuss some special cases of the obtained results. We expect that the ideas and techniques of this article will inspire interested readers working in this field. This is the main motivation of the article.

2. Results and Discussions

In this section, we discuss our main results. First of all, let us give the definition of strongly -polynomial preinvex function of higher order.

Definition 8. Let and be a nonempty invex set with respect to . A nonnegative function is said to be strongly -polynomial preinvex function of higher order, if Note that if , then the class of strongly -polynomial preinvex function of higher order reduces to the class of strongly -polynomial preinvex function which is also new class. If we consider , then the class of strongly -polynomial preinvex function of higher order reduces to the class of -polynomial preinvex functions which is also new in the literature. Similarly, if we take , then we have new class of strongly -polynomial convex function of higher order. If we take , then the above class reduces to simple strongly preinvex function of higher order. If we take , then we have strongly -polynomial preinvex function of higher order: In order to obtain following result, we need the famous Condition C which was introduced by Mohan and Neogy [55].

Condition C. Let be an invex set with respect to bifunction . Then, for any and , Note that for every , , and from Condition C, we have We now give our first main result.

Theorem 9. Let be an strongly -polynomial preinvex function with and . If satisfies Condition C, then

Proof. Since it is given that is an an strongly -polynomial preinvex function and satisfies Condition C, then Multiplying the above inequality by and then integrating the above inequality with respect to on yields Now Similarly This imples For the proof of right-hand side inequality, Multiplying the above inequality by , then integrating the resulting inequality with respect to over , we obtain This implies that Combining (19) and (22) completes the proof.

Note that if we take in Theorem 9, then we get the Hermite-Hadamard like inequality involving strongly preinvex functions.

We now derive a new auxiliary result which will be helpful in obtaining the next results of the article.

Lemma 10. Let be a differentiable function and Then, for any the following equality for -fractional integrals holds:

Proof. It suffices to show that Integrating by parts Similarly, Using (25) and (26) in (24) completes the proof.

Theorem 11. Let be a differentiable function on with and If is a strongly -polynomial preinvex function of higher order, then where