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

Abstract

A new fractional integral identity is obtained involving -th order differentiable functions and -Riemann–Liouville fractional integrals. Then, some associated estimates of upper bounds involving -preinvex functions are obtained. In order to relate some unrelated results, several special cases are discussed.

1. Introduction and Preliminaries

A set is said to be convex, if

A function 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. The concept of invex sets is defined as follows.

A set is said to be invex with respect to bi-function , if

Remark 1. If , then the concept of invex sets reduces to classical convexity. Thus, it is true that every convex set is also an invex set with respect to , but the converse is not necessarily true. For more details, see [3].

Weir and Mond [4] introd3uced the concept of preinvex functions as follows.

A function is said to be preinvex with respect to bi-function , if

Note that if we take , then from the class of preinvex functions, we recapture the class of classical convex functions.

Having inspiration from the research work of [5–7], Awan et al. [8] introduced the notion of -preinvex functions.

Let be a real function. A function is said to be -preinvex, if

It has been observed that the class of -preinvex functions generalizes several other classes of preinvexity and convexity. For example:(1)If we take , then we have the class of classical preinvex function.(2)If we take , then we have the definition of -preinvex function (see [6]).(3)If we take where , then we have the class of -preinvex functions of Breckner type (see [6]).(4)If we take , then we have the class of -Godunova–Levin–Dragomir type of preinvex functions (see [9]).(5)If we take , then we have the definition of -preinvex functions (see [8]).

It is obvious that if we take in the above discussed special cases, then we can recapture the classes of classical convexity.

Theory 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. The 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 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 -preinvex functions. The 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. The 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 -times differentiable convex functions involving Riemann–Liouville fractional integrals. Wu et al. [14] obtained fractional analogues of -th order differentiable functions involving Riemann–Liouville integrals via higher order strongly -preinvex functions. Zhang et al. [15] obtained new -fractional integral inequalities containing multiple parameters via generalized -preinvexity. The aim of this paper is to derive a new integral identity involving -times differentiable functions and -Riemann–Liouville fractional integrals. Some associated estimates of upper bounds involving -preinvex functions are also obtained. In order to relate some unrelated results, several special cases are discussed. This 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. These preliminaries will be helpful during the study of this paper.

Definition 1. (see [16]). Let. The Riemann–Liouville fractional integralsandof orderwithare defined byrespectively, and is the gamma function. Also, we define .

Mobeen and Habibullah extended the notion of Riemann–Liouville fractional integrals and introduced the concept of -Riemann–Liouville fractional integrals.

Definition 2. (see [17]). Let. The-Riemann–Liouville fractional integralsandof order withare given as follows:respectively, whereis the -gamma function (see also [18]).
Note that when -Riemann–Liouville fractional integral reduces to classical Riemann–Liouville fractional integral [16].
-beta function is defined as

2. Auxiliary Result

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

Lemma 1. Letandbe-times differentiable function. Ifand, thenwhere

Proof. We prove this result by using the mathematical induction principle. The case for is obvious. Suppose Lemma 1 holds for , that is,Now, we prove (11) for . Integrating by parts, we haveThis completes the proof.

Remark 2. (i)If we take in (11), then we have the following identity:(ii)If we take 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 to (11), then we havewhere

3. Main Results

We now derive our main results of the paper.

Theorem 1. Letandbe-times differentiable function with. If and is a-preinvex function, then

Proof. Using Lemma 1 and the -preinvexity of , we haveThis completes the proof.

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

Corollary 1. Under the assumptions of Theorem 1, ifis classical preinvex function onwith respect to, then(ii)If , then Theorem 1 reduces to the following result for the class of -preinvex function.

Corollary 2. Under the assumptions of Theorem 1, if is-preinvex function onwith respect to , then(iii)If , then Theorem 1 reduces to the following result for the class of -preinvex functions of Breckner type.

Corollary 3. Under the assumptions of Theorem 1, ifis-preinvex function onwith respect to, then where(iv)If , then Theorem 1 reduces to the following result in the class of -Godunova–Levin preinvex function.

Corollary 4. Under the assumptions of Theorem 1, ifis-Godunova–Levin preinvex function onwith respect to, then where(v)If , then Theorem 1 reduces to the following result for the class of -preinvex function.

Corollary 5. Under the assumptions of Theorem 1, ifis-preinvex function on with respect to , then

Theorem 2. Letandbe-times differentiable function with. If andis a-preinvex function, then

Proof. Using Lemma 1, Hölder’s integral inequality, and -preinvexity of , we haveThis completes the proof.

We now discuss some special cases of Theorem 2.(i)If , then Theorem 2 reduces to the following result for the class of classical preinvex function.

Corollary 6. Under the assumptions of Theorem 2, ifis classical preinvex function onwith respect to, then(ii)If , then Theorem 2 reduces to the following result for the class of -preinvex function.

Corollary 7. Under the assumptions of Theorem 2, ifis-preinvex function onwith respect to, then(iii)If , then Theorem 2 reduces to the following result for the class of -preinvex function.

Corollary 8. Under the assumptions of Theorem 2, ifis-preinvex function on with respect to , then(iv)If , then Theorem 2 reduces to the following result for the class of -Godunova–Levin preinvex function.

Corollary 9. Under the assumptions of Theorem 2, ifis-Godunova–Levin preinvex function onwith respect to, then(v)If , then Theorem 2 reduces to the following result for the class of -preinvex function.

Corollary 10. Under the assumptions of Theorem 2, ifis-preinvex function onwith respect to, then

Theorem 3. Letandbe-times differentiable function with. Ifand is a-preinvex function, then

Proof. Using Lemma 1, power mean integral inequality, and -preinvexity of , we haveThis completes the proof.

Now we will discuss some special cases of Theorem 3.(i)If , then Theorem 3 reduces to the following result for the class of classical preinvex function.

Corollary 11. Under the assumptions of Theorem 3, ifis classical preinvex function onwith respect to, then(ii)If , then Theorem 3 reduces to the following result for the class of -preinvex function.

Corollary 12. Under the assumptions of Theorem 3, ifis-preinvex function onwith respect to, then(iii)If , then Theorem 3 reduces to the following result for the class of -preinvex function.

Corollary 13. Under the assumptions of Theorem 3, ifis-preinvex function onwith respect to, then where , and are given by (25), (26), (27), and (28), respectively.(iv)If , then Theorem 3 reduces to the following result for the class of -Godunova–Levin preinvex function.

Corollary 14. Under the assumptions of Theorem 3, ifis-Godunova–Levin preinvex function onwith respect to, then where , and are given by (30), (31), (32), and (33), respectively.(v)If , then Theorem 3 reduces to the following result for the class of -preinvex function.

Corollary 15. Under the assumptions of Theorem 3, ifis-preinvex function onwith respect to, then

4. Applications

In this section, we present some applications to special means. We now recall some special means for two different positive real numbers :(i)The arithmetic mean: .(ii)The logarithmic mean: .(iii)The -logarithmic mean: .

Proposition 1. For , and , the following inequality holds:

Proof. Taking , and in inequality (24), we haveNow for the function , is a -convex function. This completes the proof.

Proposition 2. For , and , the following inequality holds:where .

Proof. Taking , and in inequality (39), we haveNow for the function , is a -convex function. This completes the proof.

Proposition 3. For, and, the following inequality holds:where .

Proof. Taking , and in inequality (46), we haveNow for the function , is a -convex function. This completes the proof.

5. Conclusion

We have derived a new fractional integral identity by using the -fractional integral and -times differentiable functions. Using this identity as an auxiliary result, we have obtained some new associated upper bounds involving -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

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was supported by HEC Pakistan under project no. 8081/Punjab/NRPU/R&D/HEC/2017.