New developments in fractional integral inequalities via convexity with applications

1.   Introduction

An important inequality for classical convex functions which has been extensively studied in recent decades is the Hermite-Hadamard's inequality, which was obtained by Hermite and Hadamard independently. This inequality gives lower and upper estimates for the integral average of any convex function formed on a compact interval encompassing the domain midpoint and endpoints. To, more precise, In ^[1] Let $f:I\subset \mathbb{R}\rightarrow \mathbb{R}$ be a convex function on the interval $I$ of real numbers and $\alpha _{1}, \alpha _{2}\in I$ with $\alpha _{1} < \alpha _{2}.$ Then

In the field of analysis, numerous mathematicians have observed the significance of the double inequality and have also used it in various useful applications. Moreover, it has been extended to various structures utilizing the classical convex function. Fractional calculus has applications in a variety of engineering and science domains, including electromagnetic, photoelasticity, fluid mechanics, electrochemistry, biological population models, optics, and signal processing. Due to its vast variety of applications, many mathematicians employed fractional calculus concepts and studied in various areas, one of which is integral inequalities for different classes of functions. For example, some authors, ^[2,3,4,5] obtained the inequalities for Riemann-Liouville fractional integrals and AB-fractional integral operator. Dragomir et al. ^[6], proved some Hermite-Hadamard type inequalities for Riemann-Liouville fractional integrals. In ^[7], Dragomir used the generalized form of Riemann-Liouville fractional integrals and proved some new Ostrowski type inequalities for bounded functions. In ^[8], he used the generalized form of Riemann-Liouville fractional integrals and proved some new trapezoid type inequalities for bounded functions. He gave trapezoid and ostrowski type inequalities using the fractional integrals. Iqbal et al. ^[9] presented some fractional midpoint type inequalities for convex functions. In recent years much attention has been devoted to the theory of convex sets and theory of convex functions by generalizing and extending these concepts in different dimensions using innovative techniques. Here, we recall the important definitions related to convex function and left-right Riemann-Liouville fractional integrals.

Definition 1.1. ^[1] Let $I$ be an interval in $\mathbb{R}.$ A function $f:I\subset \mathbb{R}\rightarrow \mathbb{R}$ is said to be convex on $\left[ \alpha _{1}, \alpha _{2}\right]$, with $\alpha _{1} < \alpha _{2},$ where $\alpha _{1}, \alpha _{2}\in I,$ if,

Definition 1.2. ^[10] For $f\in L[\alpha _{1}, \alpha _{2}].$ The left-sided and right-sided Riemann-Liouville fractional integrals of order $\kappa \in \mathbb{R }^{+}$ that are defined by

and

respectively, where $\Gamma (.)$ is Gamma function and its definition is $\Gamma (\kappa) = \int_{0}^{\infty }e^{-u}u^{\kappa -1}du.$ It is to be noted that $J_{\left(\alpha _{1}\right) ^{+}}^{0}$$f(x) = J_{\left(\alpha _{2}\right) ^{\_}}^{\kappa }f(x) = f(x).$

If we put $\kappa = 1,$ the fractional integral becomes the classical integral. The recent results and the properties concerning this operator can be found ^[11,12].

An inequality which is notable as Simpson's inequality in ^[1]:

Theorem 1.1. Suppose $f:\left[ \alpha _{1}, \alpha _{2}\right] \ \rightarrow \mathbb{R}$ is four times continuously differentiable function on $\left(\alpha _{1}, \alpha _{2}\right)$ and $\left\Vert f^{\left(4\right) }\right\Vert _{\infty } = \sup_{\theta \in \left(\alpha _{1}, \alpha _{2}\right) }\left\vert f^{\left(4\right) }\left(\theta \right) \right\vert < \infty,$ then the following inequality holds:

The accompanying ongoing improvements for Riemann-Liouville fractional integral on double and Simpson's inequalities are demonstrated by Hwang et al. (see ^[13]).

Theorem 1.2. Let $f:[\alpha _{1}, \alpha _{2}]\rightarrow \mathbb{R}$ be a differentiable function on $(\alpha _{1}, \alpha _{2})$ and $0 < \kappa \leq 1.$ If $|f^{\prime }|$ is convex function on $[\alpha _{1}, \alpha _{2}],$ then the following inequality holds:

Proposition 1.1. Suppose that all the assumptions of Theorem 2, are satisfied. If we choose $\kappa = 1$, we have trapezoid inequality:

which is obtained by Dragomir in ^[14].

Theorem 1.3. Let $f:[\alpha _{1}, \alpha _{2}]\rightarrow \mathbb{R}$ be a differentiable function on $(\alpha _{1}, \alpha _{2})$ and $0 < \kappa \leq 1.$ If $|f^{\prime }|$ is convex function on $[\alpha _{1}, \alpha _{2}],$ then the following inequality holds:

Proposition 1.2. Suppose that all the assumptions of Theorem 3, are satisfied. If we choose $\kappa = 1$, we have midpoint inequality:

which is obtained by Kirmaci in ^[15].

Theorem 1.4. Let $f:[\alpha _{1}, \alpha _{2}]\rightarrow \mathbb{R}$ be a differentiable function on $(\alpha _{1}, \alpha _{2})$ and $0 < \kappa \leq 1.$ If $|f^{\prime }|$ is convex on $[\alpha _{1}, \alpha _{2}],$ then the following inequality holds:

Proposition 1.3. Suppose that all the assumptions of Theorem 1.4, are satisfied. If we choose $\kappa = 1$, we have Simpson's inequality:

which is obtained by Sarikaya in ^[16].

In ^[17], Lian et al. presented fractional integrals inequalities for concave function as follows,

Theorem 1.5. Let $f:[\alpha _{1}, \alpha _{2}]\rightarrow \mathbb{R}$ be a differentiable function on $(\alpha _{1}, \alpha _{2})$ and $0 < \kappa \leq 1.$ If $|f^{\prime }|$ is convex function on $[\alpha _{1}, \alpha _{2}],$ then the following inequality holds:

Proposition 1.4. Suppose that all the assumptions of Theorem 1.5, are satisfied. If we choose $\kappa = 1$, we have midpoint inequality:

The current study is organized in two sections. The first section is related to the introductory body, where ideas and the hypotheses that provides the foundation for the advancement of the work has been discussed. While the second section has been divided into three sub-sections which shows the outcomes acquired for each of the inequalities under investigation. Moreover, the purpose of this paper is to study Hermite-Hadamard and Simpson's-like integral inequalities for convex functions as well as concave functions by applying the fractional concept. We also discuss the relation of our results with comparable results existing in the literature. Furthermore as applications, we find some means, $q$-digamma function and modified Bessel function type inequalities. We expect that the study initiated in this paper may inspire new research in this area.

2.   Main results

Here, we prove an important new Lemma for Riemann-Livouille fractional integrals, which plays a key role to prove our main results as follows:

Lemma 2.1. Let $f:[\alpha _{1}, \alpha _{2}]\rightarrow \mathbb{R}$ be a differentiable function on $(\alpha _{1}, \alpha _{2})$ with $\alpha _{1} < \alpha _{2}.$ If $f^{\prime }\in L\left[ \alpha _{1}, \alpha _{2}\right]$ with $0$ $<$ $\kappa \leq 1,$ $\lambda \in \left[ 0, 1\right]$, and $\rho, \vartheta \in \left[ 0, 1\right]$, then the following equality holds:

where

Proof. Integrating by parts successively, in order to compute each integral, one obtain □

Simple calculations analogously

Hence, by adding (2.1)–(2.4), and multiplying the resultant one by $\frac{ \alpha _{2}-\alpha _{1}}{2^{\kappa +2}},$ we obtain the resultant equality.

Theorem 2.1. Let $f:[\alpha _{1}, \alpha _{2}]\rightarrow \mathbb{R}$ be a differentiable function on $(\alpha _{1}, \alpha _{2})$ with $\alpha _{1} < \alpha _{2}.$ If $f^{\prime }\in L^{1}\left[ \alpha _{1}, \alpha _{2} \right]$ with $0$ $< \kappa \leq 1$ and $\rho, \vartheta \in \left[ 0, 1 \right]$, and $|f^{\prime }|$ is a convex on $[\alpha _{1}, \alpha _{2}],$ then the following inequality holds:

where,

Proof. From Lemma 2.1 and convexity, it follows that,□

The proof is completed.

Example 2.1. Let $[\alpha _{1}, \alpha _{2}] = [0, 1]$ and define the function $f:[0, 1]\rightarrow \mathbb{R}$ as $f\left(\lambda \right) = \lambda ^{3}+3.$ Let us consider the right-hand side of the inequality (2.5) as follows:

From the definitions of fractional integrals, the equalities

are valid. Finally, we have the following inequality:

As one can see in Figure 1, (2.6) in Example 1 shows the correctness of this inequality for all values of $\kappa \in \left(0, 1\right]$ and special choices of $\rho, \vartheta$. The Figure 1 represents the Graphical description of inequality (2.6) and their difference.

Remark 2.1. If we choose $\rho = \vartheta = 1,$ $\kappa = 1$, in Theorem 2.1, then inequality (2.5) reduces to inequality (1.3).

Remark 2.2. If we choose $\rho = \vartheta = 0,$ $\kappa = 1$, in Theorem 2.1, then inequality (2.5) reduces to inequality (1.5).

Remark 2.3. If we choose $\rho = \vartheta = \frac{1}{3},$ $\kappa = 1$, in Theorem 2.1, then inequality (2.5) reduces to inequality (1.7).

Theorem 2.2. Let $f:[\alpha _{1}, \alpha _{2}]\rightarrow \mathbb{R}$ be a differentiable function on $(\alpha _{1}, \alpha _{2})$ with $\alpha _{1} < \alpha _{2}$ with $0$ $<$ $\kappa \leq 1,$ $\lambda \in \left[ 0, 1 \right]$, and $\rho, \vartheta \in \left[ 0, 1\right].$ If $f^{\prime }\in L \left[ \alpha _{1}, \alpha _{2}\right]$ and mapping $|f^{\prime }|^{q}$ with $q\geq 1,$ is convex on $[\alpha _{1}, \alpha _{2}],$ then the following inequality holds:

where,

Proof. Utilizing the Lemma 2.1 and Power-mean inequality, we obtain □

The proof is completed.

Now we discuss the particular inequalities which generalize inequalities in classical sense.

Corollary 2.1. Suppose that all the assumptions of Theorem 2.2, are satisfied. If we choose $\rho = \vartheta = 1,$ $\kappa = 1$, the following inequality holds:

Corollary 2.2. Suppose that all the assumptions of Theorem 2.2, are satisfied. If we choose $\rho = \vartheta = 0,$ $\kappa = 1$, the following inequality holds:

Corollary 2.3. Suppose that all the assumptions of Theorem 2.2, are satisfied. If we choose $\rho = \vartheta = \frac{1}{3},$ $\kappa = 1,$ the following inequality holds:

Now we obtain some estimates of Simpson's and Hermite-Hadamard-inequalities for concavity.

Theorem 2.3. Let $f:[\alpha _{1}, \alpha _{2}]\rightarrow \mathbb{R}$ be a differentiable mapping on $(\alpha _{1}, \alpha _{2})$ with $\alpha _{1} < \alpha _{2}, 0$ $<$ $\kappa \leq 1,$ and $\rho, \vartheta \in \left[ 0, 1\right],$ $q\geq 1.$ If $|f^{\prime }|^{q}$ is concave on $[\alpha _{1}, \alpha _{2}],$ then the following inequality holds:

Proof. Using the concavity of $|f^{\prime }|^{q}$ and the power-mean inequality, we know that for $\lambda \in \left[ 0, 1\right]$,□

Hence

By the concavity and Jensen integral inequality, we have

which completes the proof.

As a special case of Theorem 2.3, we obtain the following result,

Corollary 2.4. Suppose that all the assumptions of Theorem 2.3, are satisfied. If we choose $\rho = \vartheta = \frac{1}{3},$ $\kappa = 1,$ we have Simpson's inequality:

Remark 2.4. Our inequality (2.10) is an improvement of Alomari inequality as obtained in ^[18].

2.1.   Applications to special means:

Let consider the following special means for $\alpha _{1}\neq \alpha _{2}.$

The arithmetic mean:

The logarithmic-mean:

The generalized logarithmic-mean:

Proposition 2.1. Suppose $r\in \mathbb{R}\backslash \{-1, 0\}$ and $\alpha _{1}, \alpha _{2}\in \mathbb{R}$ such that $0 < \alpha _{1} < \alpha _{2}$ with $q\geq 1$, then the following inequality holds:

Proof. The assertion follows from Corollary 1 for the function $f(x) = x^{r}$ and $r$ as specified above.□

Proposition 2.2. Suppose $q\geq 1$ and $\alpha _{1}, \alpha _{2}\in \mathbb{R},$ such that $0 < \alpha _{1} < \alpha _{2},$ then the following inequality holds:

Proof. The assertion follows from Corollary 2.2 for the function $f(x) = \frac{1}{x}$.□

2.2.   Q-digamma function

Suppose $0 < \mathbf{q} < 1,$ the $\mathbf{q}$-digamma function $\varphi _{ \mathbf{q}},$ is the $\mathbf{q}$-analogue of the digamma function $\varphi$ defined by (see ^[19,20]).

For $\mathbf{q} > 1$ and $x > 0,$ $\mathbf{q}$-digamma function $\varphi _{ \mathbf{q}}$defined by

Proposition 2.3. Suppose $\alpha _{1}, \alpha _{2}$ be real numbers such that $0 < \alpha _{1} < \alpha _{2}$, with $q\geq 1,$ then the following inequality holds:

Proof. The assertion can be obtained immediately by using Corollary 2.1 to $f(\varepsilon) = \varphi _{\mathbf{q}}(\varepsilon)$ and $\varepsilon > 0,$ $f^{{\prime }}(\varepsilon) = \varphi _{\mathbf{q}}^{{\prime } }(\varepsilon)$ is convex on $(0, +\infty).$□

Proposition 2.4. Suppose $\alpha _{1}, \alpha _{2}$ be real numbers such that $0 < \alpha _{1} < \alpha _{2}$, with $q\geq 1,$ then the following inequality holds:

Proof. The assertion can be obtained immediately by using Corollary 2.3 to $f(\varepsilon) = \varphi _{\mathbf{q}}(\varepsilon)$ and $\varepsilon > 0,$ $f^{{\prime }}(\varepsilon) = \varphi _{\mathbf{q}}^{{\prime } }(\varepsilon)$ is convex on $(0, +\infty).$□

2.3.   Modified Bessel function:

Recall the first kind of modified Bessel function $I_{\rho }$, which has the series representation (^[19], p.77).

where $x\in \mathbb{R}$ and $\rho > -1,$ while the second kind modified Bessel function $K_{\rho}$ (^[19], p.78) is usually defined as

Here, we consider the function $\Omega _{\rho }\left(x\right) :\mathbb{R} \rightarrow \lbrack 1, \infty)$ defined by

where $\Gamma$ is the Gamma function.

Proposition 2.5. Suppose $\rho > -1$ and $0 < \alpha _{1} < \alpha _{2}.$ Then

Specifically, if $\rho = -\frac{1}{2},$ then

Proof. The assertion can be obtained immediately by using Corollary 2.2 to $f(\varepsilon) = \Omega _{\rho }(\varepsilon),$ $\varepsilon > 0,$ and $\ \Omega _{\rho }^{{\prime }}\left(\varepsilon \right) = \frac{\varepsilon }{ \rho +1}\Omega _{\rho +1}\left(\varepsilon \right).$ Now taking into account the relations $\Omega _{-\frac{1}{2}}\left(\varepsilon \right) = \cosh \left(\varepsilon \right)$ and $\Omega _{\frac{1}{2}}\left(\varepsilon \right) = \frac{\sinh \left(\varepsilon \right) }{\varepsilon }.$□

3.   Conclusions

In this article, we have established an integral identity via Riemann-Livouille fractional integral. Based on this identity, we present several midpoint, trapezoid and Simpson's-type inequalities whose absolute values are convex and concave. It is also shown that several results are given by special cases of the main results. We deduce that the findings proved in this work are naturally universal, contribute to the theory of inequalities. Finally, we have presented some applications to special means, $q$-digamma and modifies Bessel functions with respect to our deduced results. In future studies, researchers can obtain generalized versions of our results by utilizing other kinds of convex function classes or different types of generalized fractional integral operators.

Acknowledgments

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Conflict of interest

There is no conflict of interest among the authors.