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Li, Shasha; Farid, Ghulam; Rehman, Atiq Ur; Yasmeen, Hafsa

doi:https://doi.org/10.1155/2021/2555974

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Innovative Applications of Fractional Calculus

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Volume 2021 | Article ID 2555974 | https://doi.org/10.1155/2021/2555974

Fractional Versions of Hadamard-Type Inequalities for Strongly Exponentially -Convex Functions

Shasha Li,¹Ghulam Farid,²Atiq Ur Rehman,²and Hafsa Yasmeen²

Academic Editor: Ahmet Ocak Akdemir

Received24 Apr 2021

Accepted11 Jun 2021

Published11 Aug 2021

Abstract

In this article, we prove some fractional versions of Hadamard-type inequalities for strongly exponentially -convex functions via generalized Riemann–Liouville fractional integrals. The outcomes of this paper provide inequalities of strongly convex, strongly -convex, strongly -convex, strongly -convex, strongly -convex, strongly -convex, strongly -convex, strongly exponentially convex, strongly exponentially -convex, strongly exponentially -convex, strongly exponentially -convex, strongly exponentially -convex, and exponentially -convex functions. The error estimations are also studied by applying two fractional integral identities.

1. Introduction

Fractional calculus is the study of derivatives and integrals of any arbitrary real or complex order. It is the generalization of ordinary calculus in which operations are mainly focused on integers. Its history starts when Leibniz and l’Hospital discussed the meaning of fractional order in 1695. This is the first discussion of fractional calculus. Many mathematicians devoted their efforts to make the foundation of fractional calculus. At that time, it was considered only in mathematics but now it has several applications in Science and Engineering, signal processing, mathematical biology, and rheology. In mathematics, many fractional integral operators have been introduced by researchers, see [1, 2]. Using these fractional operators, extensive inequalities are established for different types of convexity, see [3–5] and reference therein. The convex function is defined as follows:

A function , where is an interval in , is called the convex function if the following inequality holds:

Hadamard inequality is geometrical interpretation of the convex function, and it is stated as follows:

This inequality gives estimates of the mean value of a convex function. Recently, many mathematicians investigated different versions of Hadamard inequality and discussed its basic properties with corresponding fractional integral operators (see [6–9] and reference therein). Our aim is to establish Hadamard inequalities for the strongly exponentially -convex function via generalized Riemann–Liouville fractional integrals. Also, we have obtained error estimations for this convexity by using two fractional integral identities. Now, we recall the definition of the strongly exponentially -convex function.

Definition 1. (see [10]). Let be an interval containing , and let be a nonnegative function. A function is called a strongly exponentially -convex function if is nonnegative and for all , , , and , with modulus , one hasThe above definition provides some kinds of exponential convexities as follows:

Remark 1. (i)If we substitute and , then the exponentially -convex function in the second sense introduced by Qiang et al. in [11] can be obtained(ii)If we substitute , , and , then the exponentially -convex function introduced by Mehreen et al. in [12] can be obtained(iii)If we substitute , , and , then the exponentially convex function introduced by Awan et al. in [13] can be obtainedThe classical Riemann–Liouville fractional integrals are given as follows:

Definition 2. (see [14]). Let . Then, left-sided and right-sided Riemann–Liouville fractional integrals of a function of the order and are given bywhere denotes the real part of and .
Following two theorems are the fractional versions of Hadamard inequalities via Riemann–Liouville fractional integrals.

Theorem 1. (see [15]). Let be a positive function with and . If is a convex function on , then the following fractional integral inequality holds:with .

Theorem 2. (see [16]). Under the assumptions of Theorem 1, the following fractional integral inequality holds:with .

Following theorem is the error estimation of inequality (5).

Theorem 3. (see [15]). Let be a differentiable mapping on with . If is convex on , then the following fractional integral inequality holds:

The -analogue of the Riemann–Liouville fractional integral is defined as follows:

Definition 3. (see [17]). Let . Then, -fractional Riemann–Liouville integrals of order , where and , are defined aswhere is defined by [18]:Two -fractional versions of Hadamard inequality are given in next two theorems.

Theorem 4. (see [19]). Let be a positive function with . If is a convex function on , then the following inequality for -fractional integral holds:

Theorem 5 (see [20]). Under the assumptions of Theorem 4, the following inequality for -fractional integral holds:

The error estimation of inequality (10) is given in the following theorem.

Theorem 6. (see [19]). Let be a differentiable mapping on with . If is convex on , then the following inequality for -fractional integral holds:Now, we recall generalized Riemann–Liouville fractional integrals by a monotonically increasing function.

Definition 4. (see [21]). Let . Also let be an increasing and positive monotone function on ; further, has a continuous derivative on . Therefore, left as well as right fractional integral operators of order where of with respect to on are defined byThe -analogue of generalized Riemann–Liouville fractional integrals is defined as follows.

Definition 5. (see [22]). Let . Also let be an increasing and positive monotone function on ; further, has a continuous derivative on . Therefore, left as well as right -fractional integral operators of order where of with respect to on are defined byFor more details of fractional integrals, see [14, 23, 24]. We will utilize the following well-known hypergeometric, Beta, and incomplete Beta functions in our results [25]:In Section 2, we established Hadamard inequality for strongly exponentially -convex functions via generalized Riemann–Liouville fractional integrals. The special cases of these inequalities are associated with previously published papers. In Section 3, error estimations of fractional Hadamard inequality for strongly exponentially are obtained with the help of two fractional integral identities. The outcomes of this article are connected with already established results given in [15, 16, 19, 20, 26–37].

2. Main Results

This section is concerned with two fractional versions of Hadamard inequalities for strongly exponentially -convex functions. One of them is given in the following theorem.

Theorem 7. Let be a positive function with and . Also, suppose that is the strongly exponentially -convex function on with modulus . Then, for and , the following -fractional integral inequality holds for operators given in (14) and (15):where and

Proof. From strongly exponentially -convexity of , we haveBy setting and , , in (19), multiplying the resulting inequality with , and then integrating with respect to , we getLetNow one can see that will be increasing if and decreasing if . Therefore, from inequality (20), we can haveBy setting and in (22), we get the following inequality:Further, multiplying by and using Definition 5, we getThe above inequality leads to the first inequality of (17). Again, using strongly exponentially -convexity of , for , we haveBy integrating (25) over the interval after multiplying with , we getAgain using substitutions as considered in (22), the above inequality leads to the second inequality of (17).
In the following remark, we give the connection of inequality (17) with already established results.

Remark 2. (i)If we take , , , and as the identity function in (17), then the inequality stated in Theorem 2.1 in [29] is obtained(ii)If we take , , , , , , and as the identity function in (17), then Theorem 1 is obtained(iii)If we take , , , , and as the identity function in (17), then refinement of Theorem 1 is obtained(iv)If we take , , , , , , , and as the identity function in (17), then Hadamard inequality is obtained(v)If we take , , , , and in (17), then the inequality stated in Theorem 1 in [26] is obtained(vi)If we take , , , and in (17), then the inequality stated in Theorem 10 in [32] is obtained(vii)If we take , , , , , and in (17), then the inequality stated in Theorem 2.1 in [34] is obtained(viii)If we take , , , , , and as the identity function in (17), then the inequality stated in Theorem 2.1 in [31] is obtained(ix)If we take , , , and as the identity function in (17), then the inequality stated in Theorem 2 in [36] is obtained(x)If we take , , , and in (17), then the inequality stated in Corollary 1 in [35] is obtained(xi)If we take and in (17), then the inequality stated in Theorem 4 in [37] is obtained(xii)If we take , , and in (17), then the inequality stated in Corollary 1 in [37] is obtained(xiii)If we take in (17), then the inequality stated in Theorem 7 in [38] is obtainedNow, we give inequality (17) for strongly exponentially -convex, strongly exponentially -convex, strongly exponentially -convex, and strongly exponentially convex functions.

Corollary 1. If we take in (17), then the following inequality holds for strongly exponentially -convex functions:

Corollary 2. If we take and in (17), then the following inequality holds for strongly exponentially -convex functions:

Corollary 3. If we take and in (17), then the following inequality holds for strongly exponentially -convex functions:

Corollary 4. If we take , , and in (17), then the following inequality holds for strongly exponentially convex functions:

The next theorem is another version of Hadamard inequality for strongly exponentially -convex functions.

Theorem 8. Under the assumptions of Theorem 7, the following -fractional integral inequality holds:where and

Proof. By setting and , , in (19), multiplying the resulting inequality with , and then integrating with respect to , we getLetNow one can see that will be increasing if and decreasing if . Therefore, from inequality (33), we can haveBy setting and in (35), we get the following inequality:Further, multiplying above inequality by and using Definition 5, we getThe above inequality leads to the first inequality of (31). Again using strongly exponentially -convexity of , for , we haveBy integrating (38) over after multiplying with , the following inequality holds:Again, using substitutions as considered in (35), the above inequality leads to the second inequality of (31).
In the following remark, we give the connection of inequality (31) with already established results.

Remark 3. (i)If we take and in (31), then the inequality stated in Theorem 5 in [37] is obtained(ii)If we take and in (31), then the inequality stated in Corollary 3 in [37] is obtained(iii)If we take , , , , , , and as the identity function in (31), then Theorem 2 is obtained(iv)If we take , , , , , and as the identity function in inequality (31), then refinement of Theorem 2 is obtained(v)If we take , , , , , , and as the identity function in (31), then the Hadamard inequality is obtained(vi)If we take , , and in (31), then the inequality stated in Theorem 11 in [32] is obtained(vii)If we take , , , , , and as the identity function in (31), then the inequality stated in Theorem 2.1 in [30] is obtained(viii)If we take , , , and in (31), then the inequality stated in Corollary 3 in [35] is obtained(ix)If we take in (31), then the inequality stated in Theorem 8 in [38] is obtainedNow, we give inequality (31) for strongly exponentially -convex, strongly exponentially -convex, strongly exponentially -convex, and strongly exponentially convex functions.

Corollary 5. If we take in (31), then the following inequality holds for strongly exponentially -convex functions:

Corollary 6. If we take and in (31), then the following inequality holds for strongly exponentially -convex functions:

Corollary 7. If we take and in (31), then the following inequality holds for strongly exponentially -convex functions:

Corollary 8. If we take , , and in (31), then the following inequality holds for strongly exponentially convex functions:

3. Error Estimations of Hadamard Inequalities for Strongly Exponentially -Convex Functions

This section deals with error bounds of Hadamard inequalities for strongly exponentially -convex functions using generalized Riemann–Liouville fractional integrals. Estimations obtained here provide refinements of several well-known inequalities for different types of convexity. The following identity is used to prove the next theorem.

Lemma 1. (see [26]). Let and be a differentiable mapping on . Also, suppose that . Then, for , the following identity holds for the operators given in (14) and (15):

Theorem 9. Let be a differentiable mapping on such that . Also, suppose that is strongly exponentially -convex on . Then, for and , the following -fractional integral inequality holds for the operators given in (14) and (15):where .

Proof. From Lemma 1, it follows thatBy using strongly exponentially -convexity of and for , we haveUsing (47) in (46), we getIn the following remark, we give the connection of inequality (45) with already established results.

Remark 4. (i)If we take and in (45), then the inequality stated in Theorem 6 in [37] is obtained(ii)If we take and in (45), then the inequality stated in Corollary 7 in [37] is obtained(iii)If we take , , , and in (45), then the inequality stated in Theorem 12 in [32] is obtained(iv)If we take , , , , and in (45), then the inequality stated in Theorem 2 in [26] is obtained(v)If we take , , , , , and as the identity function in (45), then Theorem 6 is obtained(vi)If we take , , , , , , and as the identity function in (45), then Theorem 3 is obtained(vii)If we take , , , , , , , and as the identity function in (45), then the inequality stated in Theorem 2.2 in [27] is obtained(viii)If we take , , , and in (45), then the inequality stated in Corollary 5 in [35] is obtained(ix)If we take , , , and in (45), then the inequality stated in Theorem 12 in [32] is obtained(x)If we take in (45), then the inequality stated in Theorem 9 in [38] is obtainedNow, we give inequality (45) for strongly exponentially -convex, strongly exponentially -convex, and strongly exponentially convex functions.

Corollary 9. If we take in (45), then the following inequality holds for strongly exponentially -convex functions:

Corollary 10. If we take and in (45), then the following inequality holds for strongly exponentially -convex functions:

Corollary 11. If we take , , and in (45), then the following inequality holds for strongly exponentially convex functions:The following integral identity is useful to get our next theorem.

Lemma 2. (see [35]). Let be a differentiable mapping on such that . Then, for and , the following integral identity holds for operators given in (14) and (15):

Theorem 10. Let , , be a differentiable mapping on such that . Also suppose that is a strongly exponentially -convex function on for . Then, for and , the following fractional integral inequality holds for operators given in (14) and (15):

Proof. We divide the proof in two cases: Case 1 (for ). Applying Lemma 2 and using strongly exponentially -convexity of , we have Case 2 (for ). From Lemma 2 and using power mean inequality, we getHence, equation (53) is obtained.
In the following remark, we give the connection of inequality (53) with already established results.

Remark 5. (i)If we take in (53), then the inequality stated in Theorem 10 in [38] is obtained(ii)If we take , , , and in (53), then the inequality stated in Corollary 7 in [35] is obtained(iii)If we take , , , , , and as the identity function in (53), then the inequality stated in Theorem 2.4 in [30] is obtained(iv)If we take , , , and as the identity function in (53), then the inequality stated in Theorem 3.1 in [20] is obtained(v)If we take , , , , , , and as the identity function in (53), then the inequality stated in Theorem 5 in [16] is obtained(vi)If we take , , , , , , , and as the identity function in (53), then the inequality stated in Theorem 2.2 in [33] is obtained(vii)If we take , , , and in (53), then the inequality stated in Theorem 13 in [32] is obtained(viii)If we take and in (53), then the inequality stated in Theorem 7 in [37] is obtained(ix)If we take , , and in (53), then the inequality stated in Corollary 10 in [37] is obtainedNow, we give inequality (53) for strongly exponentially -convex, strongly exponentially -convex, strongly exponentially -convex, and strongly exponentially convex functions.

Corollary 12. If we take in (53), then the following inequality holds for strongly exponentially -convex functions:

Corollary 13. If we take and in (53), then the following inequality holds for strongly exponentially -convex functions:

Corollary 14. If we take , , and in (53), then the following inequality holds for strongly exponentially -convex functions:

Corollary 15. If we take , , , , , , and as the identity function in (53), then the following inequality is obtained:

Theorem 11. Let , , be a differentiable mapping on with . Also, suppose that is the strongly exponentially -convex function on for . Then, for and , the following fractional integral inequality holds for the operators given in (14) and (15):with .

Proof. Applying Lemma 2 and using the property of modulus, we getNow applying Hölder’s inequality for integrals, we getUsing strongly exponentially -convexity of , we getIn the following remark, we give the connection of inequality (60) with already established results.

Remark 6. (i)If we take in (60), then the inequality stated in Theorem 11 in [38] is obtained(ii)If we take and in (60), then the inequality stated in Theorem 8 in [37] is obtained(iii)If we take and in (60), then the inequality stated in Corollary 12 in [37] is obtained(iv)If we take , , , , , and as the identity function in (60), then the inequality stated in Theorem 2.7 in [30] is obtained(v)If we take , , , , , and as the identity function in (60), then the inequality stated in Theorem 3.2 in [20] is obtained(vi)If we take , , , , , , , and as the identity function in (60), then the inequality stated in Theorem 2.4 in [33] is obtained(vii)If we take , , and in (60), then the inequality stated in Corollary 9 in [35] is obtained(viii)If we take , , , and in (60), then the inequality stated in Theorem 14 in [32] is obtainedNow, we give inequality (60) for strongly exponentially -convex, strongly exponentially -convex, strongly exponentially -convex, and strongly exponentially convex functions.

Corollary 16. If we take in (60), then the following inequality holds for strongly exponentially -convex functions:

Corollary 17. If we take and in (60), then the following inequality holds for strongly exponentially -convex functions:

Corollary 18. If we take and in (60), then the following inequality holds for strongly exponentially -convex functions:

Corollary 19. If we take , , and in (60), then the following inequality holds for strongly exponentially convex functions:

4. Conclusion

In this paper, we have proved fractional versions of the Hadamard inequality and their estimations for strongly exponentially -convex functions via generalized Riemann–Liouville fractional integrals. The outcomes of this article give refinements and generalizations of fractional integral inequalities for different types of convex functions deducible from the definition of the exponentially -convex function. [39–41].

Data Availability

No data were used for the study.

Conflicts of Interest

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

Acknowledgments

The research work of the third author is supported by the Higher Education Commission of Pakistan with Project No. 7962.

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Copyright © 2021 Shasha Li 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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