Abstract

The fractional integral inequalities are crucial to deal applied problems. The present paper deals with the generalize midpoint type inequalities for a certain class of convex functions, namely, MT-convex functions in the setting of weighted fractional integral. It can be observed from the remarks given in the paper that our results or more generalized than existing results of literature and many of them can be obtained immediately from our results.

1. Introduction

Through this paper, given is a convex subset of , a continuous function is said to be convex on , if the following inequality holds: where .

Fractional calculus is one of the interesting generalizations of classical calculus, as most of the differential equations are fractional in modeling real world problems [1–3]. So the fractional integral inequalities are the interesting area of research for many researchers of analysis and inequality [4, 5]. The operates involving fractional integrals become a hot area of research for the researcher working in inequality theory (see for example [5] and references therein). Many fractional integral operators are introduced in recent years, and every operator has its own certain properties, for example, Riemann-Liouville integral operator ([6, 7]), Caputo operator [8], Hadamard operator [9], Caputo Fabrizio operator [10], and Saigo operator [11].

Like fractional calculus, fuzzy theory is also a brand new area of research. The concept of fuzzy set was developed in 1965. Since then, researchers have used this key set in many disciplines. Fuzzy set has also become an acknowledged research subject in both pure and applied mathematics and statistics, showing how this theory is highly applicable and productive in many applications [12–14]. Inequality related to fuzzy function is very papular in current years. One of the main and most powerful inequality is Hermite-Hadamard (H-H) type inequality [15], which is stated as follows:

Suppose is a convex function; then, with, .

For a further study of Hermite-Hadamard type inequality, read [16, 17].

Definition 1 (see [5, 18]). Let and be a monotone, positive and increasing function defined on having continuous derivative on . Then left sided and right sided weighted fractional integrals for , with respect to the function on , are stated as where such that

The midpoint type inequality is also a H-H type inequality, and this inequality in the setting of RL and weighted fractional integrals with positive weighted symmetric kernelbu using the mid-point of the given interval are presented in [19, 20], respectively. where is convex and continuous where is continuous and convex function and is self increasing and monotone on with is continuous function on and is an symmetric integrable function w.r.t , where [5].

This paper is organized as follows: initially, we will give some preliminaries material related to our research (Fractional Hermite-Hadamard-Fejer mid-point inequalities for a MT-Convex Function via an increasing function involving a positive weighted symmetric function). In the third section, we will present our main results.

2. Preliminaries

This section contains basic definitions and known results.

Definition 2 (see [5, 21]). Let a function . We say that is symmetric w.r.t if we have

Fejér [22] introduced the H-H-Fejer inequality by using the Definition 2, which is the weighted version of the inequality (2). where is the integrable function.

Definition 3 (see [6, 7]). and are the left- and right-sided RL fractional integrals of order defined by

The end-point inequalities of H-H type in the setting of RL fractional integrals are presented in [23, 24]. where is positive convex and continuous function on , when and .

Definition 4 (see [25]). A function is declared to be MT-convex on , if the following inequality holds: where .

Theorem 5. Let ; let be an MT-convex function and the weighted symmetric positive function w.r.t . In addition, the function is as in Definition 1; then for , we have where

Proof. The reader can prove first inequality as proved in ([20] Theorem 1).
For the second inequality, using MT-convexity of , we can get Multiply both sides by Integrating both sides with respect to over [0,1], we get From ([20] Theorem 1), we can write left hand side of the above as taking where which completes the proof. From Theorem 5, we can get following remarks as follows.

Remark 6. When , then 12 becomes where and are left- and right-weighted RL fractional integrals, respectively,

Remark 7. When and , then 12 becomes

Remark 8. When and , then 12 becomes

Remark 9. When , and , then 12 becomes

Lemma 10 (see [20] Lemma 2). Let , let be a continuous function with a derivative such that and let is as in Theorem 5 and is as in Definition 1; then, for , we have Letting implies

3. Main Results

With the help of Lemma 10, we establish H-H-F type inequalities for MT-convex functions in this section.

Theorem 11. Let ; let be a (continuously) differentiable mapping on such that and let be an integrable, positive, and weighted symmetric function with respect to . If, in addition, is MT-convex on and is an increasing and positive function from onto itself such that its derivative is continuous on , then for , the following inequalities hold: where

Proof. Using (26) and properties of modulus, we get Applying (10) on for , we get Hence, we obtain From (30) and (31), we get required inequality. From Theorem 11, we can get following remarks as follows.

Remark 12. When , then inequality (29) becomes where

Remark 13. When and , then inequality (29) becomes where

Remark 14. When , , and , then inequality (29) becomes

Theorem 15. Let ; let be a (continuously) differentiable mapping on such that and let as in Theorem 5. If is MT-convex stratifying the conditions of Definition 1, then for , we have where

Proof. By (10) on for , we get Using (26), power mean inequality and MT-convexity of from (42), (43), (44), and (45), we get the required inequality.

From Theorem 15, we can get following remarks as follows:

Remark 16. When , then inequality (41) becomes where

Remark 17. When and , then inequality (41) becomes where

Remark 18. When , and , then inequality (41) becomes

Theorem 19. Let ; let be a (continuously) differentiable mapping on such that and let be an integrable, positive, and weighted symmetric function with respect to . If, in addition, is MT-convex on with and and is an increasing and positive function from onto itself such that its derivative is continuous on , then for the following inequalities hold: where

Proof. By (10) on for , we get Using (26), Holder inequality, MT-convexity of , and properties of modulus, we get from (56); we get required inequality.

From Theorem 19, we can get following remarks as follows:

Remark 20. When , then inequality (55) becomes (37) where

Remark 21. When and , then inequality (55) becomes where

Remark 22. When , and , then inequality (55) becomes where

4. Conclusions

In this paper, the generalized midpoint type inequalities for MT-convex functions in the setting of weighted fractional integral have been established. The results established in this paper extend and generalize many existing results which are explained in the remarks. Our results can be extended for more generalized class of convex and concave functions.

Data Availability

All data required for this research are included within this paper.

Conflicts of Interest

The authors declare that they do not have any conflict of interests.

Authors’ Contributions

Yeliang Xiao supervised this work, Ahsan Fareed Shah proved the main results, Tariq Javed Zia analyzed the results and wrote the final version of the paper, and Ebenezer Bonyah wrote the first version of the paper.

Acknowledgments

This research is supported by the Department of Mathematics, University of Okara.