Abstract

In the present research, we generalize the midpoint inequalities for strongly convex functions in weighted fractional integral settings. Our results generalize many existing results and can be considered as extension of existing results.

1. Introduction

One of the most interesting research areas of classical analysis is the study of functions and operators, especially convex functions, due to its applications in both integration and differentiation. In the last few years, a great effort has been put to develop new inequalities in convex analysis to deal with the various new applications, since modern problems are modeled by fractional calculus and new applications. So, the classical convexity [1, 2] and its related [3, 4] inequalities are not enough to tackle these ones. The new fractional integral inequalities in convexity are always appreciable. Moreover, the generalized and new mode of convexity is the area of interest for most of researcher of convex analysis [5, 6].

A function is said to be convex on , if the inequality holds for all r, s and ,

Many problems may discuss in convexity of sets and functions. In recent year, convexity of sets and functions has been main object of study [7–9]. Some new generalized ideas in this point of view are pseudoconvex function, strongly convex function, quasiconvex, generalized convex function, preinvex functions [10], B-convex function, and invex functions. There are many different fundamental books of convex analysis optimization [11, 12].

Fractional calculus [13, 14] is not a new concept in mathematics, and similar discussion and controversy are observe in history by famous mathematician like Jensen, Hermite, H older, and Stolz. However, the subject of fractional calculus from an applied point of view got rapid development last years. Like other fields of mathematics, this also influences the integral inequalities and convex analysis ([15]). As a result, various trends in the result are settled recently. The famous fractional integral operators involve Riemann-Liouville [16], Caputo [17, 18], Hadamard [19], and Caputo Fabrizio [20, 21]. For more details about fractional integral operators, we refer [22–24].

The classical Hermite–Hadamard inequality is one of the most well-established inequalities in the theory of convex functions with geometrical interpretation, and it has many applications [25–27]. Recall Hermite-Hadamard-type inequality (simply H-H type inequality) which is given as:

Suppose function is convex, and then the inequality is called the Hermite-Hadamard Inequality.

In the present research, we generalize the midpoint inequalities for strongly convex functions in weighted fractional integral settings. Our results generalize many existing results and can be considered as extension of existing results.

2. Definitions and Basic Results

Definition 1. Assume that is an interval and that “a” is a positive integer. If a function is strongly convex with modulus , it is called strongly convex with modulus . for all and .

Adamek expanded on the idea of a strongly convex function. They replaced the nonnegative term with a real-valued nonnegative function and defined it as follows:

If a function is strongly convex, it is defined as such.

for all and . See [5, 28, 29] and references therein for more detail about strongly convex functionality.

Definition 2 [30]. Let be a function, Then, we say is symmetric with respect to if With the help of above definition, in [31], Fejér gave, namely, the Hermite-Hadamard-Fejér type inequality. where is the integrable function.

Definition 3. Suppose and are the left- and rigt-sided RL fractional integrals of order defined by [32] Endpoint inequalities were found, namely, the generalized and reformulated forms of H-H and H-H-F inequalities in terms of RL fractional integrals, respectively, in [22, 24]. where is the positive convex function, continuous on the closed interval when and .

Definition 4. Let and be an increasing positive and monotone function on the interval with a continuous derivative on the open interval . Then, left- and right-sided weighted fractional integral of a function , according to another function on , is defined by [25]: where such that .

Midpoint inequalities were found, namely, the generalized and reformulated forms of H-H and H-H-F inequalities in terms of RL fractional integrals and weighted fractional integrals with positive weighted symmetric function in a kernel, due to using the midpoint of the interval given by, respectively, in [33, 34].

where is convex and continuous, and

where is convex and continuous, is the monotone increasing function from onto itself with being continuous on , and is an integrable function, which is symmetric with respect to , where

The fractional integral operators induced a new diversity in inequality theory. There are many fractional integral operators introduced by different mathematicians having their own characteristics.

Lemma 5. Assume that is an integrable function and symmetric with respect to , , then (i)For each , we have(ii)For , we have

Theorem 6. Let , and let be an strong convex function and be an integrable, positive, and weighted symmetric function w.r.t . If, in addition, is an increasing and positive function from onto itself such that its derivative is continuous on , then for , the following inequalities are valid:

Proof. The strong convexity of “s” on gives So, for and , , it follows that Multiplying both sides of above equation (16) by and integrating the resulting inequality with respect to l over [0, 1] yield that From the left hand side of inequality in equation (16), we use (13) to obtain which follows that From the right hand side of inequality in (16), we use (13) to obtain By making use of (20) and (21) in (18), we get the desired result.
On the other hand, we can prove the second inequality of Theorem 6 by making use of the strong convexity of “s” to get Multiplying both sides of above equation (16) by and integrating the resulting inequality with respect to l over [0, 1] get Now, by using and (21) in (23), we get This ends our proof.

Remark 7. From Theorem 6, we can obtain some special cases as follows: (1)If , , then inequality (14) becomeswhere

Lemma 8. Let , let be a continuous with a derivative such that , and let be an integrable, positive, and weighted symmetric function with respect to . If is a continuous increasing mapping from the interval onto itself with a derivative which is continuous on , then for , the following equality is valid:

3. Main Results

In this section, by using Lemma 8, one can extend to some new H-H-F type inequalities for strong convex functions.

Theorem 9. 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 strong 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:

Proof. By making use of Lemma 8 and properties of the modulus, we obtain Since is strongly convex on , we get for : Hence, we obtain By putting value of integration in above inequalities, and simple computations yield which complete our proof.

Remark 10. From Theorem 9, we can obtain some special cases as follows: (1)If , then inequlaity (28) 3.1 becomes(2)If , , and ,then inequality 3.1 becomes(3)If , , , and v =1, then inequality 3.1 becomes

Theorem 10. 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 strong convex on with , and “s” is an increasing and positive function from onto itself such that its derivative is continuous on , then for , the following inequalities hold:

Proof. Since is strong convex on , we get for By making use Lemma 8, power mean inequality, and strong convexity of , we get Hence, the proof is completed.

Remark 11. From Theorem 10, we can obtain some special cases as follows: (1)Theorem 4 of [32] is obtained if we take, , and in Theorem 10(2)If , , then inequality (37) becomes(3)If , , and , then inequality (37) becomeswhich is already obtained in ((39) [Theorem 5]) (4)If a =0, , and v =1, then inequality (37) 3.10 become

Theorem 12. Let and 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 strong convex on with and , and “s” is an increasing and positive function from onto itself such that its derivative is continuous on , then for , the following inequalities hold:

Proof. Sinces is strong convex on , we get for By making use Lemma 8, Hölder inequality, and strong convexity of and properties of modulus, we have Hence, the proof is completed.

Remark 13. From Theorem 12, we can obtain some special cases as follows: (1)Theorem 4 of [32] is obtained if we take , , and in Theorem 12(2)If , , inequality (43) becomes(3)If , , and , then inequality (43) becomeswhich is already obtained in ((39) [Theorem 6]) (4)If , , , and , then inequality (43) becomeswhich is already obtained.

4. Conclusions

In this paper, we established the midpoint type inequalities for the strong convex function by using positive weighted symmetry kernels. As an application, our established inequalities can be applied to the special means of real numbers. Our results can be used to estimate error for the midpoint formula. It is interesting to establish midpoint type inequalities for the strong convex function in the setting of a different version of fractional integral operators.

Data Availability

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

Conflicts of Interest

The authors declare that they do not have any competing interests.

Authors’ Contributions

Hengxiao Qi designed the problem, Waqas Nazeer proved the main results, Sami Ullah Zakir wrote the first version of the paper, and Kamsing Nonlaopon analyzed the results, wrote the final version of the paper, and arranged funding for this paper.

Acknowledgments

This work was sponsored by the Innovative Engineering Scientific Research Supportive Project of the Communist Party School of the Shandong Provincial CCP Committee (Shandong Administration College) (2021cx035) and the Innovative Project of the Shandong Administrative College.