New Simpson type inequalities for twice differentiable functions via generalized fractional integrals

1.   Introduction

It is well known that Simpson's inequality is used in several branches of mathematics in the literature. For four times continuously differentiable functions, the classical Simpson's inequality is expressed as follows:

Theorem 1. Let $\digamma :\left[ a, b\right] \rightarrow \mathbb{R}$denote a four times continuously differentiable mapping on $\left(a, b\right),$ and let $\left \Vert \digamma ^{\left(4\right) }\right \Vert_{\infty } = \underset{x\in \left(a, b\right) }{\sup }\left \vert \digamma^{\left(4\right) }(x)\right \vert < \infty.$ Then, the following inequalityholds:

The convex theory is an available way to solve a large number of problems from various branches of mathematics. Hence, many authors have researched on the results of Simpson-type for convex functions. More precisely, some inequalities of Simpson's type for $s$-convex functions is proved by using differentiable functions ^[1]. In the paper ^[2], it is investigated the new variants of Simpson's type inequalities based on differentiable convex mapping. For more information about Simpson type inequalities for various convex classes, we refer the reader to Refs. ^[3,4,5,6,7] and the references therein.

In the papers ^[8] and ^[9], it is extended the Simpson inequalities for differentiable functions to Riemann-Liouville fractional integrals. Thus, several paper focused on fractional Simpson and other fractional integral inequalities for various fractional integral operators ^{[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]}. For further information about to Simpson type inequalities, we refer the reader to Refs. ^{[26,27,28,29,30,31,32]} and the references therein. In the paper ^[33], Sarikaya et al. investigated several Simpson type inequalities for functions whose second derivatives are convex.

The first and second results on fractional Simpson inequality for twice differentiable functions were established in ^[34] and ^[35], respectively. With the help of these articles, the aim of this paper is to extend the results of given in ^[33] for twice differentiable functions to generalized fractional integrals. The general structure of the paper consists of four chapters including an introduction. The remaining part of the paper proceeds as follows: In Section 2, after giving a general literature survey and definition of generalized fractional integral operators, we give an equality for twice differentiable functions involving generalized fractional integrals. In Section 3, for utilizing this equality, it is considered several Simpson type inequalities for mapping whose second derivatives are convex. In the last section, some conclusions and further directions of research are discussed.

The generalized fractional integrals were introduced by Sarikaya and Ertuǧral as follows:

Definition 1. ^[36] Let us note that a function $\varphi :[0, \infty)\rightarrow \lbrack 0, \infty)$ satisfies the following condition:

We consider the following left-sided and right-sided generalized fractional integral operators

and

respectively.

The most significant feature of generalized fractional integrals is that they generalize some types of fractional integrals such as Riemann-Liouville fractional integral, $k$-Riemann-Liouville fractional integral, Hadamard fractional integrals, Katugampola fractional integrals, conformable fractional integral, etc. These important special cases of the integral operators (1.1) and (1.2) are mentioned as follows:

1) Let us consider $\varphi \left(t\right) = t.$ Then, the operators (1.1) and (1.2) reduce to the Riemann integral.

2) If we choose $\varphi \left(t\right) = \frac{t^{\alpha }}{\Gamma \left(\alpha \right) }$ and $\alpha > 0,$ then the operators (1.1) and (1.2) reduce to the Riemann-Liouville fractional integrals $J_{_{a+}}^{\alpha }\digamma (x)$ and $J_{b-}^{\alpha }\digamma (x)$, respectively. Here, $\Gamma$ is Gamma function.

3) For $\varphi \left(t\right) = \frac{1}{k\Gamma _{k}\left(\alpha \right) }t^{\frac{\alpha }{k}}$ and $\alpha, k > 0,$ the operators (1.1) and (1.2) reduce to the $k$-Riemann-Liouville fractional integrals $J_{_{a+, k}}^{\alpha }\digamma (x)$ and $J_{b-, k}^{\alpha }\digamma (x)$, respectively. Here, $\Gamma _{k}$ is $k$-Gamma function.

In recent years, several papers have devoted to obtain inequalities for generalized fractional integrals ^{[37,38,39,40,41,42,43]}.

The first result on fractional Simpson inequality for twice differentiable functions was proved by Budak et al. in ^[34] as follows:

Theorem 2. Suppose $\digamma :[a, b]\rightarrow \mathbb{R}$ is an twice differentiable mapping $(a, b)$ so that $\digamma ^{\prime\prime }\in L_{1}\left(\left[ a, b\right] \right)$. Suppose also themapping $\left \vert \digamma ^{\prime \prime }\right \vert$ is convex on $\left[ a, b\right].$ Then, we have the following inequality

Here,

The other version of fractional Simpson inequality for twice differentiable functions was proved in ^[35] as follows:

Lemma 1. ^[35] Let us consider the function $\varpi :\left[ 0, 1\right] \rightarrow \mathbb{R}$ by $\varpi (t) = \frac{1-2\alpha }{3}+\frac{2\left(\alpha +1\right) }{3}t-t^{\alpha +1}$ with $\alpha > 0.$

1) If $0 < \alpha \leq \frac{1}{2},$ then we have

2) If $\alpha > \frac{1}{2},$ then there exist a real number $c_{\alpha }$such that $0 < c_{\alpha } < 1$ and we obtain the following equality

Theorem 3. ^[35] Assume that $\digamma :[a, b]\rightarrow \mathbb{R}$ is an absolutely continuous mapping $(a, b)$ such that $\digamma ^{\prime\prime }\in L_{1}\left(\left[ a, b\right] \right).$ Assume also that themapping $\left \vert \digamma ^{\prime \prime }\right \vert$ is convex on $\left[ a, b\right]$. Then, we have the following inequality

Here, $\Omega _{1}(\alpha)$ is defined by

2.   Some equalities for twice differentiable functions

In this section, we give an identity on twice differentiable functions for using the main results.

Lemma 2. Let $\digamma :[a, b]\rightarrow \mathbb{R}$ be an absolutely continuous mapping $(a, b)$ such that $\digamma ^{\prime\prime }\in L_{1}\left(\left[ a, b\right] \right).$ Then, the followingequality

is valid. Here, $\Omega \left(t\right) = \int \limits_{0}^{t}\Upsilon \left(s\right) ds$ and $\Upsilon \left(s\right) = \int \limits_{0}^{s}\frac{\varphi \left(\left(\frac{b-a}{2}\right) u\right) }{u}du.$

Proof. By using integration by parts, we obtain

With help of the Eq (2.2) and using the change of the variable $x = \frac{1+t}{2}b+\frac{1-t}{2}a$ for $t\in \left[ 0, 1\right],$ it can be rewritten as follows

Similarly, we get

From Eqs (2.3) and (2.4), we have

Multiplying the both sides of (2.5) by $\frac{\left(b-a\right) ^{2} }{8\Upsilon \left(1\right) }$, we obtain Eq (2.1). This ends the proof of Lemma 2.

3.   New Simpson's type inequalities for twice differentiable functions

In this section, we establish several Simpson type inequalities for mapping whose second derivatives are convex.

Theorem 4. Let us consider that the assumptions of Lemma 2 arevalid. Let us also consider that the mapping $\left \vert \digamma ^{\prime\prime }\right \vert$ is convex on $\left[ a, b\right]$. Then, we get thefollowing inequality

where $\Psi _{1}^{\varphi }$ is defined by

Proof. By taking modulus in Lemma 2, we have

By using convexity of $\left \vert \digamma ^{\prime \prime }\right \vert$, we obtain

This finishes the proof of Theorem 4.

Remark 1. If we choose $\varphi \left(t\right) = t$ in Theorem 4, then Theorem 4 reduces to [33,Theorem 2.2].

Remark 2. Let us consider $\varphi \left(t\right) = \frac{t^{\alpha }}{\Gamma \left(\alpha \right) }$ in Theorem 4. Then, the inequality (3.1) reduces to the inequality (1.3).

Corollary 1. If we assign $\varphi \left(t\right) = \frac{1}{k\Gamma _{k}\left(\alpha\right) }t^{\frac{\alpha }{k}}$ in Theorem 4, then there exist areal number $c_{\alpha }^{k}$ so that $0 < c_{\alpha }^{k} < 1$ and thefollowing inequality holds:

Here, $\Theta _{1}(\alpha, k)$ is defined by

Theorem 5. Let us note that the assumptions of Lemma 2 hold. Ifthe mapping $\left \vert \digamma ^{\prime \prime }\right \vert ^{q}$, $q > 1$is convex on $[a, b]$, then we have the following inequality

Here, $\frac{1}{p}+\frac{1}{q} = 1$ and $\Psi_{\varphi }\left(p\right)$ is defined by

Proof. By using the Hölder inequality in inequality (3.2), we obtain

With the help of the convexity of $\left \vert \digamma ^{\prime \prime }\right \vert ^{q}$, we get

This completes the proof of Theorem 5.

Remark 3. Consider $\varphi \left(t\right) = t$ in Theorem 5. Then, Theorem 5 reduces to [35,Corollary 1].

Remark 4. If it is chosen $\varphi \left(t\right) = \frac{t^{\alpha }}{\Gamma \left(\alpha \right) }$ in Theorem 5, then Theorem 5 reduces to [35,Theorem 4].

Corollary 2. Let us consider $\varphi \left(t\right) = \frac{t^{\frac{\alpha }{k}}}{k\Gamma _{k}\left(\alpha \right) }$ in Theorem 5. Then, we have

where

Theorem 6. Let us note that the assumptions of Lemma 2 hold. Ifthe mapping $\left \vert \digamma ^{\prime \prime }\right \vert ^{q}$, $q\geq 1$ is convex on $[a, b],$ then we have the following inequality

Here, $\Psi _{2}^{\varphi }$ is defined by in Theorem 4 and $\Psi _{2}^{\varphi }$is defined by

Proof. By applying power-mean inequality in (3.2), we obtain

Since $\left \vert \digamma ^{\prime \prime }\right \vert ^{q}$ is convex, we have

and similarly

Then, we obtain the desired result of Theorem 6.

Remark 5. If we take $\varphi \left(t\right) = t$ in Theorem 6, then Theorem 6 reduces to [33,Theorem 2.5].

Remark 6. Let us consider $\varphi \left(t\right) = \frac{t^{\alpha }}{\Gamma \left(\alpha \right) }$ in Theorem 6. Then, Theorem 6 reduces to [35,Theorem 5].

Corollary 3. If we choose $\varphi \left(t\right) = \frac{1}{k\Gamma _{k}\left(\alpha\right) }t^{\frac{\alpha }{k}}$ in Theorem 6, then there exist areal number $\sigma _{\alpha }^{k}$ so that $0 < \sigma _{\alpha }^{k} < 1$ andwe have the inequality

Here, $\Theta _{1}(\alpha, k)$ is defined as in (3.3) and $\Theta_{2}(\alpha, k)$ is defined by

4.   Conclusions

Fractional versions of Simpson inequalities for differentiable convex functions are investigated extensively. On the other hand, Simpson type inequalities for twice differentiable functions are also considered slightly. Hence, Simpson type inequality for twice differentiable functions by generalized fractional integrals are established in this paper. Furthermore, we prove that our results generalize the inequalities obtained by Sarikaya et al. ^[33] and Hezenci et al. ^[35]. In the future studies, authors can try to generalize our results by utilizing different kind of convex function classes or other type fractional integral operators.

Acknowledgments

This research was supported by Key Projects of Educational Commission of Hubei Province of China (D20192501), and Philosophy and Social Sciences of Educational Commission of Hubei Province of China (20Y109).

Conflicts of interest

The authors declare that they have no competing interests.