Simpson’s type integral inequalities for ĸ-fractional integrals and their applications

1.   Introduction

The following inequality is named the Simpson's integral inequality:

where $\varphi:[\varepsilon, \zeta]\rightarrow\mathbb{R}$ is a four-order differentiable mapping on $(\varepsilon, \zeta)$ and $\|\varphi^{(4)}\|_{ \infty} = \sup_{t\in(\varepsilon, \zeta)}|\varphi^{(4)}(t)| < \infty$.

Considering the Simpson-type inequality via mappings of different classes, many results involving the ordinary integrals can be found in ^{[1,3,4,5,18,19]} and the references therein.

Definition 1.1. A set $\Omega\subseteq\mathbb{R}^{n}$ is named as invex set with respect to the mapping $\eta:\Omega\times\Omega\rightarrow\mathbb{R}^{n},$ if

holds, for all $\varepsilon, \zeta\in \Omega$ and $\theta\in[0, 1].$

Definition 1.2. A mapping $\varphi:\Omega\rightarrow\mathbb{R}$ is called preinvex respecting $\eta$, if

holds for all $\varepsilon, \zeta\in\Omega$ and $\theta\in[0, 1].$

The preinvex function is an important substantive generalization of the convex function. For the properties, applications, integral inequalities and other aspects of preinvex functions, see ^{[5,7,8,9,10,11,12,13,14]} and the references therein.

Definition 1.3. Let $s\in(0, 1)$ and a mapping $\varphi:\Omega\rightarrow\mathbb{R}$ is called $s$-preinvex respecting $\eta$, if

holds for all $\varepsilon, \zeta\in\Omega$ and $\theta\in[0, 1].$

Fractional calculus has recently been the focus of mathematics and many related sciences. In addition to defining new integral-derivative operators and their properties, many researchers have achieved important results in applied mathematics, engineering and statistics. It is appropriate for the reader to review articles ^[2,15,16,17] for some recent studies. We end this section by reciting a well known $\kappa$-fractional integral operators in the literature.

Definition 1.4. ([6]) Let $\varphi\in L[\varepsilon, \zeta],$ the $\kappa$-fractional integrals $J_{\varepsilon^{+}}^{\alpha, \kappa}$ and $J_{\zeta^{-}}^{\alpha, \kappa}$ of order $\alpha > 0$ are defined by

and

respectively, where $\kappa > 0,$ and $\Gamma_{\kappa}$ is the $\kappa$-gamma function defined as $\Gamma_{\kappa}(x): = \int\limits_{0}^{\infty} \theta^{x-1}e^{-\frac{\theta^{\kappa}}{\kappa}}d\theta, \quad \Re(x) > 0,$ with the properties $\Gamma_{\kappa}(x+\kappa) = x\Gamma_{\kappa}(x)$ and $\Gamma_{\kappa}(\kappa) = 1.$

This paper aims to obtain estimation type results of Simpson-type inequality related to $\kappa$- fractional integral operators. Next, we derive the results with the boundedness of the derivative and with a Lipschitzian condition for the derivative of the considered mapping to derive integral inequalities with new bounds. Application of our results to random variables are also provided.

2.   Main results

Throughout of this work, let $\Omega\subseteq\mathbb{R}$ be an open subset respecting $\eta:\Omega\times\Omega\rightarrow\mathbb{R}\setminus\{0\}$ and $\varepsilon, \zeta\in\Omega$ with $\varepsilon < \zeta.$

To prove our results, we obtain a new integral identity as following:

Lemma 2.1. Let $\varphi :\Omega = [\varepsilon +\eta (\zeta, \varepsilon)]\rightarrow \mathbb{R}$ be a differentiable function on $(\varepsilon +\eta (\zeta, \varepsilon))$ with $\eta (\zeta, \varepsilon) > 0.$ If $\varphi ^{\prime }\in L[\varepsilon, \varepsilon +\eta (\zeta, \varepsilon)], \, \, n\geq 0$ and $\alpha, \kappa > 0,$ then the following identity holds:

where

Proof. Integration by parts, one can have

and

By adding $I_{1}$ and $I_{2}$ and multiplying the both sides $\frac{ \eta(\zeta, \varepsilon)}{2(n+1)}$, we can write

Using the fact that

we get the result.

Remark 2.1. If we choose $\eta(\zeta, \varepsilon) = \zeta-\varepsilon$ with $\kappa = 1$, then under the assumption of Lemma 2.1 one has Lemma 2.1 in ^[19].

Theorem 2.2. Let $\varphi :\Omega = [\varepsilon, \varepsilon +\eta (\zeta, \varepsilon)]\rightarrow \mathbb{R}$ be a differentiable function on $\Omega.$ If $\varphi ^{\prime }\in L[\varepsilon, \varepsilon +\eta (\zeta, \varepsilon)]$and $|\varphi ^{\prime }|^{\lambda }$ for $\lambda > 1$ with $\mu ^{-1}+\lambda ^{-1} = 1$ is $s$-preinvex function, then the following inequality holds for fractional integrals with $\alpha, \kappa > 0,$ then the following integral inequality holds:

Proof. From the integral identity given in Lemma 2.1, the H$\ddot{o}$lder integral inequality and the $s$-preinvexity of $|\varphi ^{\prime }(x)|^{\lambda },$ we have

Using the fact that $1-\chi ^{s}\leq (1-\chi)^{s}\leq 2^{1-s}-\chi ^{s}$ for $\chi \in \lbrack 0, 1]$ with $s\in (0, 1],$ we have

and

Remark 2.2. If we choose $\eta(\zeta, \varepsilon) = \zeta-\varepsilon$ with $\kappa = 1,$ then under the assumption of Theorem 2.2, one has Theorem 2.3 in ^[19].

Corollary 2.1. If we choose $\alpha = \kappa = 1,$ and $n = s = 1,$ then under the assumption of Theoremm 2.2 we have

Corollary 2.2. If we choose $\eta(\zeta, \varepsilon) = \zeta-\varepsilon$ with $\alpha = \kappa = 1,$ and $n = s = 1,$ then under the assumption of Theoremm 2.2 we have

Proof. The proof of the last inequality is obtained by using the fact that

for $0\leq j\leq1, \, \, \omega_{1}, \omega_{2}, \omega_{3}, ..., \omega_{n}\geq0; \, \, \nu_{1}, \nu_{2}, \nu_{3}, ..., \nu_{n}\geq0.$

Theorem 2.3. Let $\varphi :\Omega = [\varepsilon, \varepsilon +\eta (\zeta, \varepsilon)]\rightarrow \mathbb{R}$ be a differentiable function on $\Omega.$ If $\varphi ^{\prime }\in L[\varepsilon, \varepsilon +\eta (\zeta, \varepsilon)]$and $|\varphi ^{\prime }|$ is $s$-preinvex function, then the following inequality holds for fractional integrals with $\alpha, \kappa > 0,$ then the following integral inequality holds:

where

and

Proof. From Lemma 2.1 and using the $s$-preinvexity of $|\varphi ^{\prime }(x)|,$ we have

From (2.7), we have

By taking into account

and

Theorem 2.4. Let $\varphi :\Omega = [\varepsilon, \varepsilon +\eta (\zeta, \varepsilon)]\rightarrow \mathbb{R}$ be a differentiable function on $\Omega.$ If $\varphi ^{\prime }\in L[\varepsilon, \varepsilon +\eta (\zeta, \varepsilon)]$and $|\varphi ^{\prime }|$ is $s$-preinvex function, then the following inequality holds for fractional integrals with $\alpha, \kappa > 0,$ then the following integral inequality holds:

where

Proof. By using Lemma 2.1 and the H$\ddot{o}$lder's integral inequality for $\lambda > 1$ we have

Using $s$-preinvexity of $|\varphi ^{\prime }|,$ we get

using the fact that

A combination of (2.9)–(2.11) with (2.14) into (2.13) gives the desired inequality (2.12). The proof is completed.

Corollary 2.3. If we take $n = s = 1$ and $\alpha = \kappa = 1,$ then under the assumption of Theorem 2.4, we have

Corollary 2.4. If we take $\eta(\zeta, \varepsilon) = \zeta-\varepsilon$ with $n = s = 1$ and $\alpha = \kappa = 1,$ then under the assumption of Theorem 2.4, we have

Remark 2.3. If we take $\eta(\zeta, \varepsilon) = \zeta-\varepsilon$ with $\lambda = 1$ and $\kappa = s = 1,$ then Theorem 2.4 reduces to Theorem 2.2 in ^[19].

Remark 2.4. If we take $\eta(\zeta, \varepsilon) = \zeta-\varepsilon$ with $\lambda = 1,$ $\kappa = s = 1,$ and $\alpha = n = 1$ then Theorem 2.4 reduces to Corollary 1 in ^[18].

For deriving further results, we deal with the boundedness and the Lipschitzian condition of $\varphi^{\prime}.$

Theorem 2.5. Let $\varphi :\Omega = [\varepsilon, \varepsilon +\eta (\zeta, \varepsilon)]\rightarrow \mathbb{R}$ be a differentiable function on $\Omega.$ If $|\varphi ^{\prime }|$ is $s$-preinvex function, there exists constants $m < \mathcal{M}$ satisfying that $\infty < m\leq \varphi ^{\prime }(x)\leq \mathcal{M} < \infty$ for all $x\in \lbrack \varepsilon, \varepsilon +\eta (\zeta, \varepsilon)]$ and $\alpha, \kappa > 0,$ and $n\geq 0,$ then the following integral inequality holds:

where $\Delta _{0}$ given in (2.14).

Proof. From Lemma 2.1, we have

Using the fact that $m-\frac{m+\mathcal{M}}{2}\leq \varphi ^{\prime }\Big(\varepsilon +\frac{1-\theta }{n+1}\eta (\zeta, \varepsilon)\Big)-\frac{m+ \mathcal{M}}{2}\leq \mathcal{M}-\frac{m+\mathcal{M}}{2},$ one has

similarly,

Inequality (2.16) implies that

The proof is completed.

Corollary 2.5. If we take $\eta (\zeta, \varepsilon) = \zeta -\varepsilon$ with $\alpha = \kappa = 1,$ then under the assumption of Theorem 2.5, we have

Theorem 2.6. Let $\varphi :\Omega = [\varepsilon, \varepsilon +\eta (\zeta, \varepsilon)]\rightarrow \mathbb{R}$ be a differentiable function on $\Omega.$ If $\varphi ^{\prime }$ satisfies Lipchitz conditions on $[\varepsilon, \varepsilon +\eta (\zeta, \varepsilon)]$ for certain $\mathfrak{L} > 0$, with $\alpha > 0, \kappa > 0, \, \, n\geq 0,$ then the following inequality holds:

where $\Delta _{0}$ given in (2.14).

Proof. From Lemma 2.1, we have

Since $\varphi^{\prime}$ is Lipschitz condition on $[\varepsilon, \varepsilon+\eta(\zeta, \varepsilon)],$ for certain $\mathfrak{L} > 0,$ we have

Thus

The proof is completed.

Corollary 2.6. If we take $\eta(\zeta, \varepsilon) = \zeta-\varepsilon$ with $\alpha = \kappa = 1,$ then under the assumption of Theorem 2.6, we have

3.   Applications

3.1.   $\mathcal{F}$-divergence measure

Let the set $\psi$ and the $\delta$-finite measure $\varrho$ be given, and let the set of all probability densities on $\varrho$ to be defined on $\Lambda: = \{z|z:\psi\rightarrow\mathbb{R}, z(x) > 0, \int_{\psi}z(x)d \varrho(x) = 1\}.$

Let $\mathcal{F}:(0, \infty)\rightarrow\mathbb{R}$ be given mapping and consider $\mathcal{D}_{\mathcal{F}}(z, w)$ be defined by

If $\mathcal{F}$ is convex, then (3.1) is called as the Csiszar $\mathcal{F}$-divergence.

Consider the following Hermite-Hadamard divergence

where $\mathcal{F}$ is convex on $(0, \infty)$ with $\mathcal{F}(1) = 0.$ Note that $\mathcal{D}^{\mathcal{F}}_{HH}(z, w)\geq0$ with the equality holds if and only if $w = z.$

Proposition 3.1. Suppose all assumptions of Corollary 2.2 hold for $(0, \infty)$ and $\mathcal{F}(1) = 0,$ if $w, z\in \Lambda,$ then the following inequality holds:

Proof. Let $\Theta _{1} = \{x\in \psi :z(x) = w(x)\}.$ $\Theta _{2} = \{x\in \psi :z(x) < w(x)\}$ and $\Theta _{3} = \{x\in \psi :z(x) > w(x)\}$ and Obviously, if $x\in \Theta _{1},$ then equality holds in (3.3).

Now if $x\in \Theta _{2},$ then using Corollary 2.2 for $\varepsilon = \frac{z(x)}{w(x)}$ and $\zeta = 1,$ multiplying both sides of the obtained results by $w(x)$ and then integrating over $\Theta _{2},$ we obtain

Similarly if $x\in \Theta _{3},$ then using Corollary 2.2 for $\varepsilon = 1$ and $\zeta = \frac{z(x)}{w(x)},$ multiplying both sides of the obtained results by $w(x)$ and then integrating over $\Theta _{3},$ we obtain

Adding inequalities (3.4) and (3.5) and then utilizing triangular inequality, we get the result.

Proposition 3.2. Suppose all assumptions of Corollary 2.4 holds for $(0, \infty)$ and $f(1) = 0,$ if $w, z\in \Lambda,$ then the following inequality holds:

Proof. The proof is similar as one has done for Corollary 2.2.

3.2.   Probability density functions

Let $\mathfrak{G}:[\varepsilon, \varepsilon+\eta(\zeta, \varepsilon)] \rightarrow[0, 1]$ be the probability density function of a continuous random variable $Y$ with the cumulative distribution function of $\mathfrak{G}$

As we know $E(y) = \int\limits_{\varepsilon}^{\varepsilon+\eta(\zeta, \varepsilon)}tdF(t) = \big(\varepsilon+\eta(\zeta, \varepsilon)\big)- \int\limits_{\varepsilon}^{\varepsilon+\eta(\zeta, \varepsilon)}F(t)$

Proposition 3.3. By Corollary 2.1, we get the inequality

Proposition 3.4. By Corollary 2.3, we get the inequality

Same applications can be found for Corollary 2.2 and Corollary 2.4, respectively. We leave it to the interested readers.

4.   Conclusion

In this paper, we have established several Simpson's type inequalities via $\kappa$-fractional integrals in terms of preinvex functions. We also have obtained the inequalities applied to $\mathcal{F}$-divergence measures and application for probability density functions. These results can be viewed as refinement and significant improvements of the previously known for ^[18,19] and preinvex functions. Applications can be provided in terms of the obtained results to special means. The ideas and techniques of this paper may be attracted to interested readers.

Conflict of interest

The authors declare that there is no conflicts of interest in this paper.