On Opial-Type Inequalities for Superquadratic Functions and Applications in Fractional Calculus

Nazeer, Waqas; Farid, Ghulam; Salleh, Zabidin; Bibi, Ayesha

doi:https://doi.org/10.1155/2021/6379883

Mathematical Problems in Engineering

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Volume 2021 | Article ID 6379883 | https://doi.org/10.1155/2021/6379883

On Opial-Type Inequalities for Superquadratic Functions and Applications in Fractional Calculus

Waqas Nazeer,¹Ghulam Farid,²Zabidin Salleh,³and Ayesha Bibi²

Academic Editor: Junesang choi

Received22 May 2021

Accepted15 Jun 2021

Published30 Jun 2021

Abstract

We have studied the Opial-type inequalities for superquadratic functions proved for arbitrary kernels. These are estimated by applying mean value theorems. Furthermore, by analyzing specific functions, the fractional integral and fractional derivative inequalities are obtained.

1. Introduction and Preliminary Results

Opial’s inequality is very important and useful in the study of differential and difference equations. It was introduced by Opial in 1960. It is stated in the following theorem.

Theorem 1. (see [1]). Let be such that and for . Then,Here, is the best possible constant.

Many authors have been working continuously on Opial’s inequality and succeeded to establish very interesting results. For its numerous generalizations and extensions, we refer the readers to [2–14] and references therein. Opial-type inequalities are useful in the study of difference and differential equations, for example, in the uniqueness of initial value problems, in the existence and uniqueness of boundary value problems, and setting upper bounds of their solutions. For a historical survey of the inequalities given after the publication of inequality (1), one can study the book by Agarwal and Pang; see [2]. Mitrinović and Pečarić, in 1988, proved Opial-type inequalities for convex functions with respect to the power function by defining two classes of functions involving general kernels; see [15, 16]. Motivated by these inequalities, recently, we have studied such inequalities for convex functions by considering a generalized class of functions with arbitrary kernel [17]. Also, for special kernels, fractional Opial-type inequalities are proved for different fractional integral and derivative operators [18, 19].

Our aim in this paper is to study Opial-type inequalities for superquadratic functions which are also connected with inequalities that hold for convex functions. The established results estimate the Opial-type inequalities with the help of some suitably defined functions. The definitions and results needful for the presentation of results of this paper are given as follows.

Definition 1. (see [4]). A function is said to be convex if for all and all ,holds, where is an interval in .

Definition 2. (see [20]). A function is called a superquadratic function provided for all , there exists a real constant such thatThe following lemma explores convexity from the superquadratic function.

Lemma 1. (see [20]). Let be a superquadratic function with as in Definition 2. Then, we have(i)(ii) where is differentiable for and (iii) is convex and if Let denote the class of functions having representationwhere is a continuous function and is an arbitrary nonnegative kernel such that for and implies for every . Let denote the class of functions having representationwhere is a continuous function and is an arbitrary nonnegative kernel such that for and implies for every .

Theorem 2. (see [15, 16]). Let be a differentiable function such that, for , the function is convex and . Let where and . Then,

The reverse of the above inequality holds when is concave.

A similar result for the class is given as follows.

Theorem 3. (see [15, 16]). Let be a differentiable function such that, for , the function is convex and . Let where and . Then,

The reverse of the above inequality holds when is concave.

Recently, the following Opial-type inequalities for superquadratic functions are established.

Theorem 4. (see [21]). Let be a differentiable function such that is increasing and . Let , , and . Then, the following inequality holds for the superquadratic function :

Theorem 5. (see [21]). Under the conditions of Theorem 4, in addition, if is nonnegative, then we have

Theorem 6. (see [21]). Let be a differentiable function such that is increasing and . Let , , , and . Then, the following inequality holds for superquadratic function :

Theorem 7. (see [21]). Under the conditions of Theorem 6, in addition, if is nonnegative, then we have

The following lemmas provide classes of convex as well as superquadratic functions which will be useful to establish results of this paper.

Lemma 2. (see [22]). Let , where and for all . Then, the functions defined byare convex.

Lemma 3. (see [23]). Let , such that , for all . Consider the functions defined by

Then, the functions and are increasing. Also, if , then the functions , are superquadratic.

Lemma 4. (see [22]). Let , , , and for all . Then, the functions defined byare convex with respect to ; that is, , , are convex.

Next, we define the Riemann–Liouville fractional integral and Caputo fractional derivative as follows.

Definition 3. (see [24]). Let and . Then, the left-sided and right-sided Riemann–Liouville fractional integrals of order are defined bywhere is the gamma function.

Definition 4. (see [24]). Let and , , and . Then, the left-sided and right-sided Caputo fractional derivatives of order are defined byIn [25], Andrić et al. presented the composition identities for the left-sided and right-sided Caputo fractional derivatives; these are stated in the following two lemmas.

Lemma 5. Let , , and for ; and for . Let be such that for . Let . Then, we have

Lemma 6. Let , , and for ; and for . Let be such that for . Let . Then, we have

In the upcoming section, we prove the mean value theorems for the estimation of the nonnegative differences of the inequalities given in Theorems 4–7. In Section 3, we prove the fractional versions of the results of Section 2 for Riemann–Liouville integrals and Caputo fractional derivatives.

2. Main Results

First, we define the linear functional , from nonnegative differences of Opial-type inequalities for superquadratic functions given in Theorems 4 and 5 as follows:

Remark 1. Under the assumptions of Theorems 4 and 5, we have for .

Theorem 8. With the same assumptions as given in Theorem 4, furthermore, let be a compact interval and . Also, let where and . Then, there exists such that the following result holds:

Proof. By replacing with (defined in Lemma 3) in Theorem 4, one can have the following inequality:From inequality (21), after simplification, one can obtainSimilarly, if we take from Lemma 3 instead of in Theorem 4, then the following inequality holds:The above two inequalities lead to the following inequality:Therefore, there exists such that the following equation is obtained:which gives equation (20).

Theorem 9. With the same assumptions on as given for in Theorem 4, furthermore, if is a compact interval and where , then there exists such that the following result holds:where denominators should not be zero.

Proof. Let us define the function by , where and are given byThen, ; by applying Theorem 8 for , it follows that there exists such that we haveFrom this, one can get the required equation.

Theorem 10. With the same assumptions as given in Theorem 5, furthermore, if is a compact interval and , then there exists such that the following result holds:

Proof. By replacing with from Lemma 2 in Theorem 5, we get the following inequality:Similarly, adopting the method for functional as we did for in the proof of Theorem 8, one can see there exists such that equation (29) holds.

Theorem 11. With the same assumptions on as given for in Theorem 5, furthermore if is a compact interval and where , then there exists such that we havewhere denominators should not be zero.

Proof. Let us define the function by , where and are given byThen, ; by applying Theorem 10 for , it follows that there exists such that we haveFrom this, one can get the required equation.
Next, we define functional , from nonnegative differences of Opial-type inequalities for superquadratic functions given in Theorems 6 and 7 as follows:

Remark 2. Under the assumptions of Theorems 6 and 7, we have for .

Theorem 12. With the same assumptions of Theorem 6 on , furthermore, if is a compact interval and , then there exists such that we have

Proof. By replacing with from Lemma 3 in Theorem 6, we get the following inequality:From inequality (36), after simplification, one can obtainSimilarly, if we apply from Lemma 3 instead of in Theorem 6, then the following inequality holds:The above two inequalities lead to the following inequality:Therefore, there exists such that the following equation is obtained:which gives the required equation.

Theorem 13. With the same assumptions on as given for in Theorem 6, furthermore, if is a compact interval and where , then there exists such that we havewhere denominators should not be zero.

Proof. Let us define the function by , where and are given byThen, ; by applying Theorem 12 for , it follows that there exists such that we haveFrom this, one can get the required equation.

Theorem 14. With the same assumptions of Theorem 7 on , furthermore, if is a compact interval and , then there exists such that the following result holds:

Proof. By replacing with from Lemma 3 in Theorem 4, we get the following inequality:Rest of the proof can be followed from the proof of Theorem 12.

Theorem 15. With the same assumptions on as given for in Theorem 7, furthermore, if is a compact interval and where , then there exists such that we havewhere denominators should not be zero.

Proof. The proof can be followed from the proof of Theorem 13.

3. Fractional Versions of Mean Value Theorems

In this section, we give applications of results proved in Section 2 by choosing particular kernels. We get fractional versions of mean value theorems by applying definitions of fractional integral/derivative operators. First, we give results for Riemann–Liouville fractional integrals.

Theorem 16. With same assumptions of Theorem 4 on , furthermore, let be a compact interval and . Also, let be a Riemann–Liouville fractional integral of order , where ; then, there exists such that the following result holds:

Proof. Let us define the kernel as follows:Furthermore, we take as follows:It is clear that, for , , , . By choosing and applying Theorem 8 for the kernel given in (48), one can get the required result.

Theorem 17. With the same assumptions of Theorem 4 on , furthermore, let be a compact interval and . Also, let be a Riemann–Liouville fractional integral of order where ; then, there exists such that we havewhere denominators should not be zero.

Proof. It is easy to prove by applying Theorem 9 for the kernel defined in (48) and using the function given by (49).

Theorem 18. With the same assumptions of Theorem 5 on , furthermore, let be a compact interval and . Also, let be a Riemann–Liouville fractional integral of order where ; then, there exists such that the following result holds:

Proof. The proof is similar to the proof of Theorem 16.

Theorem 19. With the same assumptions of Theorem 5 on , furthermore, let be a compact interval and . Also, let be a Riemann–Liouville fractional integral of order where ; then, there exists such that we havewhere denominators should not be zero.

Proof. It is easy to prove by applying Theorem 11 for the kernel defined in (48) and using the function given by (49).

Theorem 20. With the same assumptions of Theorem 6 on , furthermore, let be a compact interval and . Let be a Riemann–Liouville fractional integral of order where ; then, there exists such that the following result holds:

Proof. Let us consider the kernel as defined in (48) and given in (49). If we set , then . Furthermore, is increasing for , on ; therefore, we have . By applying Theorem 12, we get the required result.

Theorem 21. With the same assumptions of Theorem 6 on , furthermore, let be a compact interval and . Furthermore, let be a Riemann–Liouville fractional integral of order . If and , then there exists such that the following equality holds:where denominators should not be zero.

Proof. It is easy to prove by applying Theorem 13 for the kernel defined in (48) and using the function given by (49).

Theorem 22. With the same assumptions of Theorem 7 on , furthermore, let be a compact interval and . Let be a Riemann–Liouville fractional integral of order . If , then there exists such that the following result holds:

Proof. The proof is similar to the proof of Theorem 20.

Theorem 23. With the same assumptions of Theorem 7 on , furthermore, let be a compact interval and . Let be a Riemann–Liouville fractional integral of order . If and , then there exists such that the following equality holds:where denominators should not be zero.

Proof. It is easy to prove by applying Theorem 15 for the kernel defined in (48) and using the function given by (49).
Next, we give the results for Caputo fractional derivatives using their composition identities.

Theorem 24. With the same assumptions of Theorem 4 on , furthermore, let be a compact interval and . Also, let and , for and such that for . Let . Then, for , the following result holds:

Proof. Let us consider the kernel as follows:Furthermore, we take as follows:It is clear that, for , , , . By choosing and applying Theorem 8 for the kernel given in (59), one can get the required result.

Theorem 25. With the same assumptions of Theorem 4 on , furthermore, let be a compact interval and . Let and , for . Also, let , , and . Then, there exists such that we havewhere denominators should not be zero.

Proof. It is easy to prove by applying Theorem 9 for the kernel defined in (58) and using function given by (59).

Theorem 26. With the same assumptions of Theorem 5 on , furthermore, let be a compact interval and . Also, let and , for , and such that for . Let . Then, the following result holds for :

Proof. The proof is similar to the proof of Theorem 24.

Theorem 27. With the same assumptions of Theorem 5 on , furthermore, let be a compact interval and . Let and , for . Also, let , , and . Then, there exists such that we havewhere denominators should not be zero.

Proof. It is easy to prove by applying Theorem 11 for the kernel defined in (58) and using function given by (59).

Theorem 28. With the same assumptions of Theorem 6 on , furthermore, let be a compact interval and . Also, let and , for , and such that for . Let . Then, for , the following result holds:

Proof. Let us consider the kernel as defined in (58) and given in (59). If we set , then . Furthermore, is increasing for , on ; therefore, we have . By applying Theorem 12, we get the required result.

Theorem 29. With the same assumptions of Theorem 6 on , furthermore, let be a compact interval and . Let and , for . Also, let , , and . Then, there exists such that, for , we havewhere denominators should not be zero.

Proof. It is easy to prove by applying Theorem 13 for the kernel defined in (58) and using function given by (59).

Theorem 30. With the same assumptions of Theorem 7 on , furthermore, let be a compact interval and . Also, let and , for , and such that for . Let . Then, for , the following result holds:

Proof. The proof is similar to the proof of Theorem 28.

Theorem 31. With the same assumptions of Theorem 7 on , furthermore, let be a compact interval and . Let and , for . Also, let and , , and . Then, there exists such that, for , we havewhere denominators should not be zero.

Proof. It is easy to prove by applying Theorem 15 for the kernel defined in (58) and using function given by (59).

Data Availability

No additional data were required for the findings of results of this paper.

Conflicts of Interest

The authors declare no conflicts of interest.

Authors’ Contributions

All the authors contributed equally to this article.

Acknowledgments

This study was partially supported by the Center of Research and Innovation Management, Universiti Malaysia Terengganu.

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Copyright © 2021 Waqas Nazeer et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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