New Developments of Hermite–Hadamard Type Inequalities via <i>s</i>-Convexity and Fractional Integrals

Khan, Khuram Ali; Fatima, Saeeda; Nosheen, Ammara; Matendo Mabela, Rostin

doi:https://doi.org/10.1155/2024/1997549

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Research Article | Open Access

Volume 2024 | Article ID 1997549 | https://doi.org/10.1155/2024/1997549

New Developments of Hermite–Hadamard Type Inequalities via s-Convexity and Fractional Integrals

Khuram Ali Khan,¹Saeeda Fatima,¹Ammara Nosheen,¹and Rostin Matendo Mabela²

Academic Editor: Kenan Yildirim

Received03 Aug 2022

Revised09 Apr 2023

Accepted18 Dec 2023

Published16 Jan 2024

Abstract

In this paper, we present an identity for differentiable functions that has played an important role in proving Hermite–Hadamard type inequalities for functions whose absolute values of first derivatives are -convex functions. Meanwhile, some Hermite–Hadamard type inequalities for the functions whose absolute values of second derivatives are -convex are also established with the help of an existing identity in literature. Many limiting results are deduced from the main results which are stated in remarks. Some applications of proved results are also discussed in the present study.

1. Introduction

Inequalities have been proved to be the most efficient tools for the construction of several branches in mathematics. In the field of classical differential and integral equations, the inequalities have played an important role [1, 2]. Charles Hermite and Jacques Hadamard derived Hermite–Hadamard inequality which is stated as follows.

1.1. Hermite–Hadamard Inequality

Let function be a convex function and with , then the following inequalities hold:

If is concave, then equation (1) holds in back direction. Barsam et al. [3] introduced the integral identities associated with Hermite–Hadamard inequality for -convex functions. Barsam and Sattarzadeh [4] found Hermite–Hadamard type inequalities involving fractional integrals for uniformly convex functions. Because of many applications of Hermite–Hadamard type inequalities [5–9] and fractional calculus [10–14], it is intended to study the Hermite–Hadamard type inequalities involving fractional integrals. Mohammed [15] obtained the inequalities via factional integrals of convex functions with respect to increasing functions. Set [16] defined new Ostrowski type inequalities via Riemann–Liouville fractional integrals for -convex functions. Abdeljawad et al. [17] introduced new Simpson-type inequalities for -convex functions. Işcan [18] introduced some inequalities for -convex functions involving fractional integrals. Usta et al. [19] introduced trapezoid type inequalities for -convex functions with generalized fractional operators. Butt et al. [20] obtained integral identity; by using that identity, new inequalities were obtained via a general form of fractional integral operators. Agarwal et al. [21] gave Hermite–Hadamard type inequalities for generalized -fractional integrals. Sahoo et al. [22] obtained integral inequalities by using -Riemann–Liouville fractional operator for -convex functions. Sarikaya et al. [23] introduced the following Hermite–Hadamard type inequalities involving Riemann–Liouville fractional integrals.

Theorem 1 (see [23]). Let be a positive function with and . If is a convex function on , then the following inequality for fractional integrals holds:

where and indicate the left-sided and right-sided Riemann–Liouville fractional integrals of the order which are as follows:respectively, and is the classical Euler gamma function.-convex functions are generalization of classical convex function. It is remarkable that Özdemir et al. [24] defined the -convex function as follows: [24]. Let be a function, then is called -convex function, if for each and , . Hölder Inequality for Integrals [25]. Let and . If and are real functions on and if are integrable functions on , , then The following power-mean integral inequality is an elementary result of Hölder inequality: Power-Mean Integral Inequality [25]. Let . If and are real functions defined on and if are integrable functions on , then

Some authors applied classical inequalities such as Hölder inequality and power mean inequality and also applied the special functions like classical Euler–gamma and beta functions to fractional integrals to get new integral inequalities for the different classes of convex functions. Qaisar et al. obtained some new Hermite–Hadamard inequalities involving fractional integrals for convex functions [14]. Some refinements for integral and sum forms of Hölder inequality were elaborated by Işcan [1]. Authors are motivated by the results given in [3, 26]. The purpose of this paper is to establish new Hermite–Hadamard type inequalities involving fractional integrals via -convex functions.

2. Hermite–Hadamard Type Inequalities Involving Fractional Integrals for the Class of Differentiable Functions

To prove our main results associated with Hermite–Hadamard type inequalities involving fractional integrals, we need the following lemma.

Lemma 2. Let be a differentiable mapping on and with . If , then the following equality for fractional integral with holds:for all .

Proof. ConsiderSubstituting in the above equation, we getNow, considerSubstituting in the above equation, we getAdding equations (9) and (11), we getThe proof is completed.

Remark 3. By replacing with in equation (9) and with in equation (11) and adding the resulting equations, we obtain the following equation:Substituting in the second term of R. H. S of the equation (13), then equation (13) becomes ([26], Lemma 1.2).

The following two examples show that the class of functions whose absolute values are differentiable -convex functions is nonempty.

Example 1. Let be defined by , in this case, the function is a –convex function for . Because if for each , we putIt is easy to see that for and , so , thereforeand that means

Example 2. Let be defined by , , and , then the function is a -convex function on the interval . Because if we putso , therefore

Theorem 4. Let and be a positive function with and . If is a -convex function on , then the following inequality for fractional integrals holds:

Proof. Applying the -convexity of , we getMultiply both sides of equation (20) by and integrate w.r.t over [0, 1].Now, substituting in the first term of L. H. S of equation (21) and in the second term of L. H. S of equation (21), we getMultiplying both sides of the above equation by , we getThe proof is completed.

Theorem 5. Let be a differentiable mapping on and with such that . If is -convex on , then the following inequality for fractional integral holds:for some fixed .

Proof. Using Lemma 2,(since is -convex)The proof is completed.

Theorem 6. Let be a differentiable mapping on and with such that . If is -convex on , then the following inequality for fractional integral holds:for some fixed .

Proof. According to Lemma 2,Applying the Hölder inequality (5) in equation (28), we get(since is -convex)The proof is completed.

Remark 7. By applying the power-mean integral inequality (6) in equation (28), then we also get the inequality (27).

Theorem 8. Let be a differentiable mapping on and with such that . If is -convex on , then the following inequality for fractional integral holds:for some fixed .

Proof. According to Lemma 2,According to Hölder inequality (5), we get(since is -convex)The proof is completed.

3. Hermite–Hadamard Type Inequalities Involving Fractional Integrals for the Class of Twice Differentiable Functions

Dragomir et al. [27] defined the following identity involving Riemann–Liouville fractional integrals.

Lemma 9 (see [27]). Let be a twice differentiable function on . Assume that with and , then the following identity for fractional integral with holds:

To prove our results associated with Hermite–Hadamard inequalities involving fractional integrals for twice differentiable functions, we need Lemma 9.

Theorem 10. Let be a twice differentiable function on such that is -convex function on . Suppose that with , , and , then the following inequality holds:

Proof. According to Lemma 9, we getApplying the -convexity of , we getSince , , and , we have .The proof is completed.

Theorem 11. Let be twice differentiable function on . Assume that and such that is -convex function on . Suppose that with and , then the following inequality holds:

Proof. According to Lemma 9, we getAccording to Hölder inequality (5), Since , and , we have .(since is -convex)Substituting equations (44)–(46) in equation (43), we getThe proof is completed.

Remark 12. For , inequality (40) becomes as follows:For , Corollary 3.6 in [5] reduces to inequality (48) and, for , Corollary 3.5 in [5] reduces to inequality (48).

3.1. Comparison

Here, we have compared Theorem 11 with Corollary 5.8 in [19].

For , Theorem 11 becomes as follows:

Let , , and .

Similarly, Corollary 5.8 in [19] for becomes as follows:

Let , , and .

The difference of bounds of Theorem 11 is 1.164508 and difference of bounds of Corollary 5.8 in [19] is 1.28683. Hence, our result is more efficient.

Theorem 13. Let be a differentiable function on . Assume that and such that is -convex function on . Suppose that , with and , then the following inequality holds:

Proof. According to Lemma 9, we getApplying the power-mean integral inequality (6), we getSimplifying(since is -convex)Since , , and , we have .The proof is completed.

Theorem 14. Suppose that is a differentiable on such that , where with . Assume that is -convex on with and , we have the following inequality:with .

Proof. According to Lemma 9, we getApplying the Hölder inequality (5), we get(since is -convex)Since , , and , we have .The proof is completed.

Theorem 15. Assume that is a differentiable on such that , where with . Assume that is -convex on with and , then the following inequality holds:with .

Proof. According to Lemma 9, we getApplying the Hölder inequality (5), we get(since is the -convex)Since , , and , we have .The proof is competed.

Remark 16. For , Theorem 10 reduces to Theorem 2 in [27] and Theorem 11 reduces to Theorem 3 in [27]. When both , then Theorem 10 reduces to Theorem 2 [28].

4. Applications to Some Special Means

Consider the following special means for arbitrary real numbers and as follows:

Proposition 17. Let , , , and , then

Proof. This statement follows from Theorem 10, by using and .

Proposition 18. Let , , , and , then

Proof. This statement follows from Theorem 11, by using , , and .

Remark 19. For and , Proposition 4.4 in [5] reduces to inequality (71).

Proposition 20. Let , , , and , then

Proof. This statement follows from Theorem 13, by using , , and .

5. Conclusion

In this paper, authors have established Hermite–Hadamard type inequalities involving Riemann–Liouville fractional integrals via -convex functions by applying two different techniques. In first part, an identity is proved in which a differentiable function is presented in the form of Riemann–Liouville fractional integrals of first derivatives of function. Furthermore, this identity is used to establish Hermite–Hadamard type inequalities in which the absolute values of first derivatives are -convex functions. In the second part, an identity in which a function in the form of integral of double derivative of function is used to establish Hermite–Hadamard inequalities in which the absolute values of second derivatives are -convex functions. The limiting cases included some existing results in the literature. Some applications of the obtained results are also described in the form of means. This method can also be applicable for other classes of convex functions.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare they have no conflicts of interest.

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Copyright © 2024 Khuram Ali Khan 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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