New Simpson Type Integral Inequalities for -Convex Functions and Their Applications

Abstract

First, we consider a new Simpson’s identity. This identity investigates our main results that consist of some integral inequalities of Simpson’s type for the –convex functions. From our main results, we obtain some special cases which are discussed in detail. Finally, some applications on the Bessel functions, special means of distinct positive real numbers, and error estimation about Simpson quadrature formula are presented to support our theoretical results.

1. Introduction and Preliminaries

Integral inequalities are generally applicable in many branches of mathematics such as mathematical analysis, fractional calculus, discrete fractional calculus, and abstract spaces; for an overview, the reader should see the literature on integral inequalities, e.g., [1–11] and the references therein.

Nowadays, the study of convexity is considered as an original icon in the investigation of theoretical behavior of mathematical inequalities (e.g., [12–15]). Recently, several works on integral inequalities for convex functions were conducted. In particular, much attention has been given to the theoretical studies of inequalities on different types of convex functions such as –geometrically convex functions [16], -convex functions [17], quasiconvex function [18], -convex function [19, 20], -convex functions [21], -convex functions [22], -convex functions [23], and a new class of convex functions [24], and many other types can be found in [25].

Now we recall some useful definitions of convex functions that are essential for our present investigation.

Definition 1. (see [25]). A function is said to be convex, ifholds for all and . Likewise, is concave if is convex.

Definition 2. (see [26]). Let be fixed. A function is said to be –convex (in the second sense), ifholds for all and .
Recently, many well-known inequalities haven been established for the convex function and its types such as Hermite–Hadamard type inequalities [27, 28], Hermite–Hadamard–Fejer type inequalities [29], Ostrowski type inequalities [30], Hardy type inequalities [31], Olsen type inequalities [32], Gagliardo–Nirenberg type inequalities [33], Opial type inequalities [34, 35], and Rozanova type inequalities [36]. However, our focus in this paper is on another type of inequality, namely, the Simpson’s inequality, which is obtained by Dragomir et al. [37], and their main result is as follows.

Theorem 1. (see [37]). Assume that is a four times continuous and differentiable function on such that with . Then, the following integral inequality holds:Just like the aforementioned inequalities, several integral inequalities related to the Simpson’s integral inequality (3) has been found for the convex functions (see, e.g., [38–45]).
In view of the above indices and literatures, we will prove a new Simpson’s integral identity. From this, then we will obtain several Simpson’s integral inequalities and some special cases.

The rest of this article is organized in the following way: Section 2 contains the main results of the article which consists of proving several Simpson’s integral inequalities and some related results. In Section 3, we give some applications of the Bessel functions, special means, and error estimation about the Simpson quadrature formula to support the main results. Section 4 is for the conclusions.

2. Main Results

We need the following Lemma in order to establish our main results.

Lemma 1. Let be an function on such that with . Also, let , and then for any positive integer number , we havewhere

Proof. From the definition of , we can deduce thatBy applying the integration by parts to the right side of (6), we haveand similarly,for all . Then, substituting (7) and (8) in the equality (6), we getBy summing over from 0 to , we can deduce the desired identity (4).

Remark 1. Taking in Lemma 1, we have the following Simpson’s rule (see [43]):where is defined by (5).

Theorem 2. Let be an function on such that with and be fixed. Also, let and be an –convex function on , and then for any positive integer number , we havewhere

Proof. By using Lemma 1, –convexity of and properties of modulus, we can deducewhich completes the proof.

Corollary 1. Choosing in Theorem 2, we get the following inequality for the convex function:where

Corollary 2. Choosing in Theorem 2, we get

Theorem 3. Let be an function on such that with and be fixed. Also, let and be an –convex function on , and then for any positive integer number , we havewhere is as defined in Theorem 2 and .

Proof. By making use of Lemma 1, Hölder’s inequality, –convexity of , and properties of modulus, we haveThis ends our proof.

Corollary 3. Choosing in Theorem 3, we get the following inequality for the convex function:

Corollary 4. Choosing in Theorem 3, we get

Theorem 4. Let be an function on such that with and be fixed. Also, let and be –convex function on , and then for any positive integer number , we havewhere for all and are as defined in Theorem 2.

Proof. By using Lemma 1, the well-known power mean inequality, –convexity of , and properties of modulus, we havewhich completes the proof.

Remark 2. Choosing in Theorem 4, we get Theorem 2

Corollary 5. Choosing in Theorem 4, we get the following inequality for the convex function:where for all are defined as in Corollary 1.

Corollary 6. Choosing in Theorem 4, we get

3. Examples and Applications

3.1. Bessel Functions

Consider the function given bywhere is the modified Bessel function of the first kind defined by (see (2) in [46], p. 77):

The first order derivative formula of is given by [46]:and the second derivative can be easily calculated from (27) to be

Example 1. Let and . Then, by applying Corollary 2 (note that all assumptions are satisfied) and the identities (27) and (28), we havewhere are as in Theorem 2.

Example 2. Let and . Then, by applying Corollary 4 (note that all assumptions are satisfied) and the identities (27) and (28), we have

Example 3. Let and . Then, by applying Corollary 6 (note that all assumptions are satisfied) and the identities (27) and (28), we havewhere are as in Theorem 2.

3.2. Special Means

We consider the following two special means for different positive real numbers and :(i)The arithmetic mean:(ii)The generalized logarithmic mean:

Proposition 1. Let and . Then, we have for a fixed where for all are as defined in Theorem 2.

Proof. By applying Theorem 2 with and , then we can obtain the desired result (34).

Proposition 2. Let and . Then, we have for a fixed where with .

Proof. By applying Theorem 3 with and , we can obtain the desired result (37) directly.

Proposition 3. Let and . Then, we have for a fixed where for all are defined as in Theorem 2 and .

Proof. By applying Theorem 4 with and , we can directly obtain the desired result (43).

3.3. Simpson Quadrature Formula

Let be the partition of the points of the interval and consider the quadrature formulawhereis the Simpson version and denotes the associated approximation error. In this section, we are going to derive some new estimates for the Simpson quadrature formula.

Proposition 4. Let , be fixed and be a differentiable function on . If and is an –convex function on , then we havewhere for all are as defined in Theorem 2 and

Proof. By applying Theorem 2 on the subintervals of the partition , we can deduceTherefore, from (41), it follows thatThis gives our Proposition 4.

Proposition 5. Let , be fixed and be a differentiable function on . If and is an –convex function on , then we have for and :where and are as defined in Proposition 4.

Proof. By the same method used for Proposition 4 and by using Theorem 3, one can prove this proposition directly.

Proposition 6. Let , be fixed and be a differentiable function on . If and is an –convex function on , then we have for :where and are as defined in Proposition 4.

Proof. By the same method used for Proposition 4 and by using Theorem 4, one can prove this proposition directly.

4. Conclusion

We have obtained a new Simpson identity at first. By using this, we have established some interesting Simpson’s type integral inequalities for –convex functions. In order to illustrate the efficiency of our main results, some applications to the Bessel functions, special means, and error estimation about Simpson quadrature formula are provided. To the best of our knowledge, these results are new in the literature. Since the class of convex (–convex) functions have large applications in many mathematical areas, they can be applied to obtain several results in convex analysis, quantum mechanics, special functions, mathematical inequalities, and related optimization theory and may stimulate further research in different areas of pure and applied sciences.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Acknowledgments

The third author would like to thank Prince Sultan University for funding this work through the research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) (group number RG-DES-2017-01-17).