Abstract
The goal of this paper is to establish some weighted Simpson type inequalities for functions whose first derivatives are convex involving Reimann–Liouville integral operators. In order to obtain our results, we first prove a new integral identity as an auxiliary result. Based on this identity we establish some fractional weighted Simpson type inequalities for functions whose modulus of the first derivatives are convex. Several special cases are discussed. Error estimates for some numerical quadrature rules are furnished.
1. Introduction
Convexity is an analytical tool, it represents a basic notion in geometry and widely used in many areas of mathematics such as optimization, calculus of variations, and graph theory. This principle has a closed relationship with theory of inequality. It is clear that the most important inequality directly related to convexity is the so-called HermiteHadamard inequality which can be stated as follows: for any convex function on the interval , we have the following double inequality:
Integral inequalities involving convex functions plays an important role in the different areas of science besides mathematics, such as physics, economics, biology, and engineering sciences, where most of the problems come down by solving integrals in the majority without difficult or impossible to solve directly which leads us to use approximate methods in other words quadrature formulas.
In the last decades, the study of error estimation of quadrature formulas has become a hot and attractive topic in the field of research. Several Newton–Cotes formulas have been studied by many researchers under the different classes of functions. The most important and remarkable three-point Newton–Cotes formula is that of Simpson which can be stated as follows:where is four times continuously differentiable function on , and .
In recent years, many researchers have studied the inequality (2), and several papers have been published dealing with refinements, generalizations, extensions as well as analogous versions of (2), for more details we refer the readers to [1–8] and references therein.
Sarikaya et al. [9] gave the following Simpson type inequalities for convex differentiable functions.
Theorem 1.1. Letbe a differentiable mapping onand letsuch thatand. Ifis convex, then we have
Theorem 1.2. Letbe a differentiable mapping onand letsuch thatand. Ifis convex wherewith, then we have
Theorem 1.3. Letbe a differentiable mapping onand letsuch thatand. Ifis convex where, then we have
Sarikaya et al. [10] proposed a refinement of the result given in Theorem 1.2 as follows:
Theorem 1.4. Letbe a differentiable mapping onletsuch thatand. Ifis convex wherewith, then we have
Recently, Kashuri et al. [11], established the weighted version of Simpson type inequalities.
Theorem 1.5. Letbe a differentiable function onsuch thatwith, and letbe continuous and symmetric function with respect to. Ifis convex, then we have
Theorem 1.6. Letbe a differentiable function onsuch thatwith, and letbe continuous and symmetric function with respect to. Ifis convex wherewith, then we have
Theorem 1.7. Letbe a differentiable function onsuch thatwith, and letbe continuous and symmetric function with respect to. Ifis convex where, then we have
Nowadays, fractional calculus has become a popular tool for scientists. It has been successfully used in the various fields of science and engineering [12]. Its main strength in describing the memory and genetic properties of the different materials and processes has aroused great interest among researchers from different fields. Regarding some papers dealing with fractional integral inequalities we advise readers to refer to some related studies [11–21].
The main goal of this study is to establish some fractional weighted Simpson type inequalities generalizing some earlier published papers by using a new identity. Several known results can be derived according to the values of the parameter or the weighted function . We end the paper by some applications in numerical integration to demonstrate the efficacy of our results.
2. Preliminaries
In this section, we recall certain definitions and a lemma that we will use in the sequel.
Definition 2.1. [22]. A function is said to be convex, ifholds for all and all .
Definition 2.2. [23]. A function is said to be symmetric with respect to , iffor all.
Definition 2.3. [18]. Let . The Riemann–Liouville integrals and of order with are defined byrespectively, where is the Gamma function and .
Definition 2.4. [18]. The beta function is defined for all complex numbers and with and as follows:
Definition 2.5. [18]. The hypergeometric function is defined for and , as follows:where is the beta function.
Lemma 2.1. [24]. For anyinand, we have
3. Main Results
Lemma 3.1. Letbe a differentiable function on, withand letbe a symmetric with respect to. If, thenwherewithand
Proof. LetandIntegrating by parts and changing the variable, we obtainSimilarly, we haveFrom the symmetry of we haveandSumming Equations (21) and (22), then multiplying the result by and using Equations (23) and (24), we get the desired result.
In what follows we assume .
Theorem 3.1. Letbe a differentiable function on, with, and letbe symmetric with respect toand. Ifis convex, then we have
Proof. Using Lemma 3.1, modulus, convexity of and Lemma 2.1, we obtainwhere we have used the facts thatandThe proof is achieved.
Corollary 3.1. In Theorem 3.1, if we takewe getwhere
Remark 3.1. In Theorem 3.1, if we take , we obtain the first inequality of Corollary 2.4 from [11]. Moreover, if we choose , we get the second inequality of Corollary 2.4 from [11].
Corollary 3.2. In Theorem 3.1, if we use the convexity ofi.e., we obtain
Corollary 3.3. In Corollary 3.2, if we takewe getwhereis defined as in Equation (32).
Remark 3.2. In Corollary 3.3, if we take we get Corollary 2.5 from [11]. Moreover, if we choose , we obtain Theorem 5 from [9].
Theorem 3.2. Letbe a differentiable function on, with, and letbe symmetric with respect toand. Ifis convex wherewith, then we havewhereis defined by Equation (15), andare the beta and hypergeometric functions, respectively.
Proof. Using Lemma 3.1, modulus, Hölder inequality, convexity of , and Lemma 2.1, we obtainwhere we have used the facts thatandthe proof is achieved.
Corollary 3.4. In Theorem 3.2, if we takewe getwhereis defined as in Equation (32).
Remark 3.3. In Theorem 3.2, if we take we get Corollary 2.8 from [11]. Moreover if we take , we obtain Corollary 3.1 from [10].
Corollary 3.5. In Theorem 3.2, if we use the convexity ofi.e., we obtain
Corollary 3.6. In Corollary 3.5, if we take.whereis defined as in Equation (32).
Remark 3.4. In Corollary 3.5, if we take , we obtain the first inequality of Corollary 2.4 from [11]. Moreover if we choose , we get Theorem 6 from [9].
Corollary 3.7. In Corollary 3.5, if we use the discrete power mean inequality, we get
Corollary 3.8. In Corollary 3.13, if we take.whereis defined as in Corollary 3.12.
Corollary 3.9. In Corollary 3.13, if we take
Corollary 3.10. In Corollary 3.13, if we takeandwe get
Theorem 3.3. Letbe a differentiable function on, with, and letbe symmetric with respect toand. Ifis convex where, then we havewhereis defined by Equation (15), andare the beta and hypergeometric functions, respectively.
Proof. Using Lemma 3.1, modulus, power mean inequality, convexity of and Lemma 2.1, we obtainwhere we have used Equations (28)–(31). The proof is achieved.
Corollary 3.11. In Theorem 3.3, if we takewe getwhereis defined as in Equation (32).
Corollary 3.12. In Theorem 3.3, if we takewe get
Corollary 3.13. In Theorem 3.3, if we takeandwe get
Corollary 3.14. In Theorem 3.3, if we use the convexity of, we obtain
Corollary 3.15. In Corollary 3.14, if we take, we obtainwhereis defined as in Equation (32).
Remark 3.5. In Corollary 3.14, if we take , we obtain Corollary 2.12 from [11]. Moreover if we choose , we get Theorem 7 from [9].
Corollary 3.16. In Corollary 3.14, if we use the discrete power mean, we obtain
Corollary 3.17. In Corollary 3.16, if we takewe getwhereis defined as in Equation (32).
Corollary 3.18. In Corollary 3.14, if we takewe get
Corollary 3.19. In Corollary 3.18, if we takeandwe get
4. Example
In this section, we utilize Matlab software to provide a graphical representation for analyzing the behavior of the estimates. The right-hand side is depicted in red color, while the left-hand side is depicted in blue color.
Example 4.1. Assume thatand. Clearly we haveand.
From Theorem 3.1, we derive the following inequality, which is illustrated in Figure 1.
5. Applications
This section will be devoted to the applications of the results obtained, knowing that in general in this type of problem the applications relate to the special averages, the random variable or in the numerical integration of which it is the center of our interest.
Let be the partition of the points of the interval , and consider the quadrature formula [11]whereand denotes the associated approximation error.
Proposition 5.1. Letandbe a differentiable function on, with, and letbe symmetric with respect toand. Ifis convex, then we havewhere.
Proof. Applying Corollary 3.9 on the subintervals for of the partition , we getSumming above inequality for all and using property of modulus, we obtain the desired result.
Proposition 5.2. Letandbe a differentiable function on, with, and letbe symmetric with respect toand. Ifis convex, then we havewhere.
Proof. Applying Corollary 3.12 on the subintervals for of the partition , we getSumming above inequality for all and using property of modulus, we obtain the desired result.
Proposition 5.3. Letandbe a differentiable function on, with, and letbe symmetric with respect toand. Ifis convex, then we havewhere.
Proof. Applying Corollary 3.18 on the subintervals for of the partition , we getSumming above inequality for all and using property of modulus, we obtain the desired result.
Now, recalling some particularly means.
The Arithmetic mean: .
The harmonic mean: with .
The logarithmic mean: .
Proposition 5.4. Letwithand let, then we have
Proof. The assertion Corollary 3.19, applied to function .
6. Conclusion
In this study, we have considered the fractional weighted Simpson type integral inequalities for functions whose first derivatives are convex. We have established a novel weighted identity that incorporates the Riemann–Liouville integral operator. We have established some new fractional weighted Simpson type inequalities. We have discussed according to the values of the parameter or the weight function some special cases. Several known results have been derived. Applications of our findings are provided. We hope that the ideas of this paper will inspire researchers working in field of inequalities to generalize our results for different kinds of classical and generalized convexity. In future work, we can expand upon our research to encompass various types of calculus, including quantum calculus as well as non-Newtonian calculus.
Data Availability
No underlying data were collected or produced in this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.