Inequalities for Estimations of Integrals Related to Higher-Order Strongly -Polynomial Preinvex Functions

Abstract

In this paper, we establish an integral identity associated with -times differentiable functions. The result is then used to derive some integral estimations for higher-order strongly -polynomial preinvex functions. Finally, we apply the obtained inequalities to construct new inequalities involving special means.

1. Introduction

The convexity properties of functions have been a powerful tool for dealing with various problems of pure and applied sciences. In recent decades, the concept of classical convexity has been extended and generalized in different directions, and the study relevant to this subject has stimulated an increased interest. In [1], Weir and Mond introduced the notion of preinvexity as follows.

Definition 1. A set is said to be invex with respect to if

Definition 2. A function is said to be preinvex with respect to ifFor , the class of preinvex functions reduces to the class of convex functions. It had been shown that the preinvex functions may not be convex functions. For example, the function is not a convex function, but it is a preinvex function with respect to provided thatMohan and Neogy [2] and Mititelu [3] discussed the properties of invex sets and preinvex functions. Noor [4] and Noor et al. [5] investigated integral inequalities of Hermite–Hadamard type for log-preinvex functions and -preinvex functions, respectively. Wu et al. [6] considered harmonically -preinvex functions and relevant inequalities. Peng et al. [7] established fractional Simpson’s inequalities with the aid of -preinvexity. Du et al. [8] dealt with integral inequalities involving generalized -convex functions on fractal sets. Saleh [9] defined -preinvex function and studied the properties of -preinvex functions. Mohammed [10] focused on the integral inequalities for preinvex functions via the generalized beta function.
Very recently, Toplu et al. [11] introduced and investigated the class of -polynomial convex functions as follows.

Definition 3. Let . A nonnegative function is called an -polynomial convex function if for any and ,Note that if we take in the above inequality, then we have 1-polynomial convexity which is just the classical convexity. It is also necessary to mention here that every nonnegative convex function is also an -polynomial convex function. For more properties on the -polynomial convex function, see [11].
Kadakal [12] and Karamardian [13] put forward the class of strongly convex functions, respectively. Strong convexity is just the strengthening version of convexity, which reads as follows.

Definition 4. A function is said to be a strongly convex function with modulus ifIn order to deal with the problems of mathematical programs with equilibrium constraints, Lin and Fukushima [14] proposed a generalization of the strongly convex function, which is called the higher-order strongly convex functions.

Definition 5. A function is said to be a strongly convex function with order and modulus ifThe higher-order strongly convex functions would reduce to the strongly convex function in the special case when .
As an extension of classical convex functions, the strongly convex functions have been widely used in establishing new inequalities and dealing with estimations of integrals. Angulo et al. [15] and Zhang et al. [16] presented some inequalities for strongly -convex functions and strongly -convex functions, respectively. In order to further generalize the strongly convex functions, recently, some researchers, such as Mishra and Sharma [17], Noor and Noor [18], and Mohsen et al. [19], began to study the higher-order strongly convex functions. Wu et al. [20, 21] gave some estimates of the upper bound for differentiable functions associated with k-fractional integrals and higher-order strongly convex functions. Khan et al. [22] and Ullah et al. [23] introduced the concept of coordinate and the technique of majorization into the study of strongly convex functions.
In this paper, we introduce the notion for a new class of strongly convex functions which are named as the strongly -polynomial preinvex functions of order .

Definition 6. Let . A nonnegative function is said to be a strongly -polynomial preinvex function of order with modulus ifholds for all and .

Remark 1. Note that(i)If we take in Definition 6, then we have the class of strongly -polynomial preinvex functions(ii)If we take in Definition 6, then we have the class of strongly -polynomial convex functions of order (iii)If we take and in Definition 6, then we have the class of strongly -polynomial convex functions(iv)If we take in Definition 6, then we have the class of strongly preinvex functions of order This shows that the class of strongly -polynomial preinvex functions of order is quite unifying one as it relates several unrelated classes of convexity.
The main objective of this paper is to establish a new integral identity for -times differentiable functions. With the help of this identity, we deduce some inequalities for estimations of integrals associated with -times differentiable strongly preinvex functions of order . As applications, we discuss some special cases of the main results and establish several new inequalities involving special means.

2. A Key Lemma

Here, we establish an auxiliary result which will play a significant role in the development of our main results.

Lemma 1. Let be an -times differentiable function on with . If , then the following integral equality holds for :

Proof. Integrating by parts, we haveSimilarly, we deduce thatCombining (9) and (10) completes the proof of Lemma 1.

3. Main Results

We are now in a position to state our main results.

Theorem 1. Let be an -times differentiable function on with and . If is a strongly -polynomial preinvex function of order with modulus , thenwhereand is the beta function.

Proof. Using Lemma 1 and the fact that is a strongly -polynomial preinvex function of order , we havewhereThe proof of Theorem 1 is completed.
In the following, we discuss some special cases of Theorem 1.(1)If we take the limit in Theorem 1, we get a result for the -polynomial preinvex function.

Corollary 1. Under the assumptions of Theorem 1, if is an -polynomial preinvex function, then

 where(2)If we take in Theorem 1, we get a result for the strongly -polynomial preinvex function.

Corollary 2. Under the assumptions of Theorem 1, if is a strongly -polynomial preinvex function, then

 where(3)If we take in Theorem 1, we get a result for the higher-order strongly preinvex function.

Corollary 3. Under the assumptions of Theorem 1, if is a strongly preinvex function of order , then

Theorem 2. Let be an -times differentiable function on with , , and . If is a strongly -polynomial preinvex function of order with modulus , thenwhere is the arithmetic mean.

Proof. Utilizing Lemma 1, Hölder’s integral inequality, and the fact that is a strongly -polynomial preinvex function of order , one hasThis completes the proof of Theorem 2.
In the following, we discuss some special cases of Theorem 2.(1)Putting in Theorem 2, we obtain a result for the -polynomial preinvex function.

Corollary 4. Under the assumptions of Theorem 2, if is an -polynomial preinvex function of order , then

(2)Putting in Theorem 2, we obtain a result for the strongly -polynomial preinvex function.

Corollary 5. Under the assumptions of Theorem 2, if is a strongly -polynomial preinvex function, then

(3)Putting in Theorem 2, we obtain a result for the higher-order strongly preinvex function.

Corollary 6. Under the assumptions of Theorem 2, if is a strongly preinvex function of order , then

Theorem 3. Let be an -times differentiable function on with , and . If is a strongly -polynomial preinvex function of order with modulus , thenwhere and are the expressions as described in Theorem 1.

Proof. Applying Lemma 1, power-mean integral inequality, and the fact that is a strongly -polynomial preinvex function of order , we haveTheorem 3 is proved.
Finally, we discuss some special cases of Theorem 3.(1)Letting in Theorem 3, we get a result for the -polynomial preinvex function.

Corollary 7. Under the assumptions of Theorem 3, if is an -polynomial preinvex function of order , then

(2)Choosing in Theorem 3, we get a result for the strongly -polynomial preinvex function.

Corollary 8. Under the assumptions of Theorem 3, if is a strongly -polynomial preinvex function, then

 (3) Choosing in Theorem 3, we get a result for the higher-order strongly preinvex function.

Corollary 9. Under the assumptions of Theorem 3, if is a strongly preinvex function of order , then

4. Application to Special Means

We consider the following special means.(1)The arithmetic mean:(2)The mean of the function on[e, f]

Proposition 1. Let be m-times differentiable function on with and . If is a strongly n-polynomial preinvex function of order with modulus , then the following inequality holds:where and are as in Theorem 1.

Proof. Taking , and using Theorem 1, we get the desired result (32).

Proposition 2. Let be m-times differentiable function on with , , and . If is a strongly n-polynomial preinvex function of order with modulus , then the following inequality holds:

Proof. Setting , and applying Theorem 2, we get the desired result (33).

Proposition 3. Let be m-times differentiable function on with , and . If is a strongly n-polynomial preinvex function of order with modulus , then the following inequality holds:where and are as in Theorem 1.

Proof. Taking , and utilizing Theorem 3, we get the desired result (34).

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Authors’ Contributions

All authors contributed equally and significantly to this paper. All authors read and approved the final manuscript.

Acknowledgments

This work was supported by the Natural Science Foundation of Fujian Province of China (Grant no. 2020J01365) and the Teaching Reform Project of Fujian Provincial Education Department (Grant no. FBJG20180120).