Abstract
The aim of this paper is to derive from an auxiliary identity some new generalizations of fractional integral inequalities of Simpson, midpoint, and trapezoid using the class of harmonic convex functions. To show that our results are quite unifying, we discuss several new special cases. Finally, some applications regarding error estimations for the Simpson quadrature formula are presented to support our theoretical results.
1. Introduction and Preliminaries
Fractional calculus was born on September 1695. Although the history of fractional calculus is very old, in recent years it has emerged as an interdisciplinary subject. Besides its great many applications in applied sciences, it has also played a significant role in modern analysis. Due to these facts, it received special attention by mathematicians, and as a result, a variety of new significant generalizations of the classical concepts of fractional calculus have been proposed in the literature (for details, see [1]). Since the emergence of fractional calculus, several researchers have started obtaining the fractional analogues of classical mathematical objects. For example, Sarikaya et al. [2] were the first to obtain some new fractional analogues of Hermite–Hadamard’s inequality. This idea opened a new direction of research for inequalities experts (for some more details, see [3–8]).
We now recall some preliminary concepts that will be needed throughout this paper.
Let be a nonempty interval of . A function is said to be convex, if
Recently, İşcan [9] introduced the class of harmonic convex functions as
Let , be a nonempty interval. A function , is said to be harmonic convex, if
In order to explain more the concept of harmonic convex functions, we state the following remark, which may be of interest to the reader.
Remark 1. (i)By virtue of the so-called arithmetic-harmonic mean inequality, namely, it is easy to see that every convex increasing function on is harmonic convex on . However, a harmonic convex function is not necessary convex. For example, the concave function is harmonic convex on .(ii)In terms of means, (2) can be simply written as With this, if we set for , it is not hard to see that (2) is equivalent to This means that, is harmonic convex on if and only if is convex on . It follows that every harmonic convex function on is continuous on .(iii)According to (4), if is harmonic convex on then we haveor, equivalently,Now, let us recall some basic notions about fractional integrals. The Riemann–Liouville fractional integrals are defined as
Definition 1 (see [1]). Let . The Riemann–Liouville fractional integrals and of order with are defined byrespectively, andis the well-known Gamma function. Also, we define
The -Riemann–Liouville fractional integrals are defined as
Definition 2 (see [10]). Let . The -Riemann–Liouville fractional integrals and of order with are given as follows:respectively, and stands for the -Gamma function.
In order to calculate integrals, we need hypergeometric functions. The integral form of the hypergeometric function is given as
Definition 3. The hypergeometric function has the following integral representation:where and refers to the Euler beta function.
Otherwise, Sarikaya and Ertugral [11] defined a new generalization of fractional integrals (which we call as Sarikaya fractional integral) as itemized in what follows.
Let be a function satisfying the following conditions:(1),(2) for , where , and are independent of .Under the assumptions of , the left-sided and the right-sided generalized fractional integrals areSarikaya’s fractional integrals are the generalization of some well-known fractional integrals like the Riemann–Liouville fractional integrals [1], -Riemann–Liouville fractional integrals [10], Katugampola fractional integrals [12], conformable fractional integrals [7], etc.(1)If we take in operators (13) and (14), we have the classical Riemann integrals.(2)If we choose in operators (13) and (14), we get the Riemann–Liouville fractional integrals (see [1]).(3)If we substitute in operators (13) and (14), we obtain the -Riemann–Liouville fractional integrals (see [10]).(4)If we take in operators (13) and (14), we have the conformable fractional integrals, which were defined by Khalil et al. in [13].(5)If we choose for , in operators (13) and (14), we get the left-sided and the right-sided fractional integrals with the exponential kernel, which were defined in [11, 12].To highlight our goal in this paper, let us recall the following results obtained by Dragomir et al. in [14]:
Theorem 1. Let us assume that is a four times differentiable function on , such that with . Then, the following integral inequality holds:
Just like the aforementioned inequalities, several integral inequalities related to Simpson’s integral inequality (15) have been found for convex functions (see [15–22]). But, our fundamental target in this paper is on another type of inequality, namely, the Simpson’s inequality for fractional integrals, by using the concept of harmonic convex functions.
The remainder of this paper will be organized as follows: in Section 2 we state an auxiliary lemma which will be a primordial tool for establishing our main results. Motivated by this lemma, we derive in Section 3 some new generalizations of fractional integral inequalities of Simpson, midpoint, and trapezoid type using a class of harmonic convex functions. To show that our results are quite unifying, we will discuss several new special cases. Finally, Section 4 displays some applications regarding error estimations of the Simpson quadrature formula as support for our theoretical results. We hope that the ideas and the techniques developed in this paper will inspire interested readers working in this field.
2. A New Auxiliary Result
In this section, we derive a key lemma that will help us in deriving our main results.
Lemma 1. Let be as in the previous section. Let be a differentiable function, with . If is integrable on , then for , we havewhereas before , and the function is defined by
Proof. First, we mention that the function defined by (17) is continuous on . By the standard rule of integration by parts, and using (13) and (14) with some algebraic operations, we haveAdding equalities (18) and (19) and multiplying by the factor , we obtain the required result.
Remark 2. (i)For the sake of simplicity, we write in the following and instead of and , respectively.(ii)The previous lemma stems its importance in the fact that it includes a large class of examples and situations. Indeed, first, the parameters and could be chosen in an arbitrary manner. Secondly, as previously mentioned, belongs to a large class of functions. And thirdly, as explained in Remark 1, could be easily chosen as a harmonic convex function. Let us explain more these latter points in what follows.(i)Taking special values for and in Lemma 1, we obtain the following:(1) If , then(2) For , we get(3) With , we have(ii)We now discuss some other variants for Lemma 1 when choosing special cases for as follows:(I) If we take , then(II) If we choose , then(III) With , we haveRemark If we take or for , in Lemma 1, we can derive new identities regarding conformable fractional integrals and fractional integrals with the exponential kernel. We left to the reader the task of formulating these identities in a detailed manner.
We also state the following lemma which will be needed in the sequel.
Lemma 2. Let and let be two continuous functions on . Then we have
Proof. It is based on the standard integral Hölder inequality when writingThe details are straightforward and therefore omitted here for the reader. □
3. Results and Discussions
3.1. Main Results
Our first main result in this section is as follows.
Theorem 2. Under the assumptions of Lemma 1, if the function is harmonic convex on , then the following inequality holds for Sarikaya fractional integralswhere
Proof. Using Lemma 1 and the properties of the modulus, we haveSince the function is harmonic convex on , with the help of (7), we getThis completes the proof.
We now state our second main result as recited in the following.
Theorem 3. Under the assumptions of Lemma 1, if the function is harmonic convex on for some , thenwhereand are defined as in Theorem 2.
Proof. Using Lemma 1, the properties of the modulus and Lemma 2, respectively, we haveSince the function is harmonic convex on , with the help of (7), we getThis completes the proof.
We end this section by stating our third main result, which reads as follows:
Theorem 4. Under the assumptions of Lemma 1, let us assume that the function is harmonic convex on for some . Let be such that . Then we havewhere is defined, for , by
Proof. Using Lemma 1, the properties of the modulus and Hölder’s inequality, we haveSince the function is harmonic convex on , with the help of (7), we getThis completes the proof.
3.2. Special Cases
We now discuss some special cases of results discussed in the main results section.
First, we consider some particular values of and in Theorem 2.(1)For , we have where(2)For , we get where(3)For , we obtainwhere
We now discuss some particular cases of Theorem 2 when choosing special functions .(I) If we take , then we have where(II)If we choose , then we get where(III)With , we obtainwhere
We now discuss some special cases of Theorem 3
First, we consider some particular values of and in Theorem 3.(1)For , we have where(2)For , we get where(3)For , we obtainwhere
We now discuss some particular cases of Theorem 3 for special functions .(I)If we take , then we have where are given by (55), (56), (57), and (58), respectively,(II)If we choose , then we get where are given by (60), (61), (62), and (63), respectively, and(III)With , we obtainwhere are given by (65), (66), (67), and (68), respectively, and
We now consider some particular values of and in Theorem 4.(1)For , we have where are given by (38), (39), (40), and (41), respectively, and(2)For , we get where are given by (38), (39), (40), and (41), respectively, and(3)For , we obtainwhere are given by (38), (39), (40), and (41), respectively, and(I)Under the assumptions of Theorem 4, if we take , then we have where is defined, for , by and are given by (38), (39), (40), and (41), respectively.(II)Under the assumptions of Theorem 4, if we choose , then we get where is defined, for , by and are given by (38), (39), (40), and (41), respectively.(III)Under the assumptions of Theorem 4, if we substitute , then we obtainwhere, for we setand are given by (38), (39), (40), and (41), respectively.
Remark 3. . If we choose or for , in Theorems 2, 3, and Theorem 4, we can establish new inequalities regarding conformable fractional integrals and fractional integrals with the exponential kernel. We omit their proofs here and the details are left to the interested reader.
4. Simpson Quadrature Formula
In this section we will present some applications of the integral inequalities obtained above, to find new error bounds for the Simpson quadrature formula. We preserve the same notations as in the previous sections.
First, we fix two parameters .
For , let be a partition of .
We set
Our aim here is to approximate the following integral by demandingwhere is the remainder term for .
Using the above notations, we are in the position to prove the following error estimations.
Proposition 1. Under the assumptions of Theorem 2, the following inequality holds:whereand
Proof. We use Theorem 2 on the subinterval of the closed interval when . Following (53) we have, for all ,Summing inequality (104) over from 0 to and using the properties of the modulus, we obtain the desired inequality.
Proposition 2. . Under the assumptions of Theorem 3, the following inequality holds:where are as in Proposition 1,and
Proof. We apply the same technique as in the proof of Proposition 1, by using Theorem 3 with and with the help of (78).
Proposition 3. . Under the assumptions of Theorem 4, the following inequality holds:where is defined, for , by
Proof. Applying the same technique as in the proof of Proposition 1 but via Theorem 4 with and using (92).
5. Conclusion
We have derived a new generalized fractional integral identity using the Sarikaya fractional integral. Using this, we have established some new associated fractional integral inequalities of the Simpson, midpoint, and trapezoid types using the class of harmonic convex functions. To show that our results are quite unifying, we have discussed several new special cases. In order to illustrate the significance of our main results, some applications regarding error estimations for the Simpson quadrature formula have been discussed.
Data Availability
No data were used to support this study.
Conflicts of Interest
The author declares that there are no conflicts of interest.