Automatic recognition of white blood cell images with memory efficient superpixel metric GNN: SMGNN

Yuanhong Jiang; Yiqing Shen; Yuguang Wang; Qiaoqiao Ding; Yuanhong Jiang; Yiqing Shen; Yuguang Wang; Qiaoqiao Ding

doi:10.3934/mbe.2024095

Mathematical Biosciences and Engineering

2024, Volume 21, Issue 2: 2163-2188. doi: 10.3934/mbe.2024095

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Research article Special Issues

Automatic recognition of white blood cell images with memory efficient superpixel metric GNN: SMGNN

1.
School of Mathematical Sciences, MOE-LSC, Shanghai Jiao Tong University, Shanghai 200030, China
2.
Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai 200030, China
3.
Department of Computer Science, Johns Hopkins University, USA
4.
Shanghai Artificial Intelligence Laboratory, Shanghai 200433, China

Academic Editor: Yang Kuang

Received: 14 October 2023 Revised: 19 December 2023 Accepted: 22 December 2023 Published: 10 January 2024

An automatic recognizing system of white blood cells can assist hematologists in the diagnosis of many diseases, where accuracy and efficiency are paramount for computer-based systems. In this paper, we presented a new image processing system to recognize the five types of white blood cells in peripheral blood with marked improvement in efficiency when juxtaposed against mainstream methods. The prevailing deep learning segmentation solutions often utilize millions of parameters to extract high-level image features and neglect the incorporation of prior domain knowledge, which consequently consumes substantial computational resources and increases the risk of overfitting, especially when limited medical image samples are available for training. To address these challenges, we proposed a novel memory-efficient strategy that exploits graph structures derived from the images. Specifically, we introduced a lightweight superpixel-based graph neural network (GNN) and broke new ground by introducing superpixel metric learning to segment nucleus and cytoplasm. Remarkably, our proposed segmentation model superpixel metric graph neural network (SMGNN) achieved state of the art segmentation performance while utilizing at most 10000 $\times$ less than the parameters compared to existing approaches. The subsequent segmentation-based cell type classification processes showed satisfactory results that such automatic recognizing algorithms are accurate and efficient to execeute in hematological laboratories. Our code is publicly available at https://github.com/jyh6681/SPXL-GNN.

Keywords:

Citation: Yuanhong Jiang, Yiqing Shen, Yuguang Wang, Qiaoqiao Ding. Automatic recognition of white blood cell images with memory efficient superpixel metric GNN: SMGNN[J]. Mathematical Biosciences and Engineering, 2024, 21(2): 2163-2188. doi: 10.3934/mbe.2024095

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Abstract

1. Introduction

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

$\begin{eqnarray} \Big\vert\frac{1}{6}\Big[\varphi(\varepsilon)+4\varphi\Big(\frac{ \varepsilon+\zeta}{2}\Big)+\varphi(\zeta)\Big]-\frac{1}{\zeta-\varepsilon} \int\limits_{\varepsilon}^{\zeta}\varphi(t)dt\Big\vert\leq\|\varphi^{(4)} \|_{\infty}(\zeta-\varepsilon)^{4}, \end{eqnarray}$

(1.1)

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

$\begin{equation*} \varepsilon+\eta(\zeta, \varepsilon)\in \Omega \end{equation*}$

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

$\begin{eqnarray*} \varphi(\varepsilon+\eta(\zeta, \varepsilon))\leq(1-\theta)\varphi( \varepsilon)+\theta\varphi(\zeta), \end{eqnarray*}$

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

$\begin{eqnarray*} \varphi(\varepsilon+\eta(\zeta, \varepsilon))\leq(1-\theta^{s})\varphi( \varepsilon)+\theta^{s}\varphi(\zeta), \end{eqnarray*}$

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

$\begin{eqnarray} J_{\varepsilon^{+}}^{\alpha, \kappa} = \frac{1}{\kappa\Gamma_{\kappa}(\alpha)} \int\limits_{\varepsilon}^{x}(x-\theta)^{\frac{\alpha}{\kappa} -1}\varphi(\theta)d\theta, \quad(0\leq\alpha \lt x \lt \zeta) \end{eqnarray}$

(1.2)

and

$\begin{eqnarray} J_{\zeta^{+}}^{\alpha, \kappa} = \frac{1}{\kappa\Gamma_{\kappa}(\alpha)} \int\limits_{x}^{\zeta}(\theta-x)^{\frac{\alpha}{\kappa}-1}\varphi(\theta)d \theta, \quad(0\leq\alpha \lt x \lt \zeta), \end{eqnarray}$

(1.3)

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:

$\begin{eqnarray} &&\Psi (\varepsilon, \zeta ;n, \alpha, \kappa) \\ &:& = \frac{\eta (\zeta, \varepsilon)}{2(n+1)}\Big[\int\limits_{0}^{1}\Big( \frac{2(1-\theta)^{\frac{\alpha }{\kappa }}-\theta ^{\frac{\alpha }{\kappa } }}{3}\Big)\varphi ^{\prime }\Big(\varepsilon +\frac{1-\theta }{n+1}\eta (\zeta, \varepsilon)\Big)d\theta \\ &&+\int\limits_{0}^{1}\Big(\frac{\theta ^{\frac{\alpha }{\kappa } }-2(1-\theta)^{\frac{\alpha }{\kappa }}}{3}\Big)\Big]\varphi ^{\prime }\Big( \varepsilon +\frac{n+\theta }{n+1}\eta (\zeta, \varepsilon)\Big)d\theta \Big], \end{eqnarray}$

(2.1)

where

$\begin{eqnarray*} &&\Psi (\varepsilon, \zeta ;n, \alpha, \kappa) \\ &:& = \frac{1}{6}\Bigg[\varphi (\varepsilon)+\varphi (\varepsilon +\eta (\zeta, \varepsilon))+2\varphi \Big(\varepsilon +\frac{1}{n+1}\eta (\zeta , \varepsilon)\Big)+2\varphi \Big(\varepsilon +\frac{n}{n+1}\eta (\zeta , \varepsilon)\Big) \\ &&\quad -\frac{\Gamma _{\kappa }(\alpha +\kappa)(n+1)^{\frac{\alpha }{ \kappa }}}{6\big(\eta (\zeta, \varepsilon)\big)^{\frac{\alpha }{\kappa }}} \Big[J_{(\varepsilon)^{+}}^{\alpha, \kappa }\varphi \Big(a+\frac{1}{n+1} \eta (\zeta, \varepsilon)\Big)\Big]+J_{\big(\varepsilon +\eta (\zeta , \varepsilon)\big)^{-}}^{\alpha, \kappa }\varphi \Big(\varepsilon +\frac{n}{ n+1}\eta (\zeta, \varepsilon)\Big)\Big] \\ &&\quad -\frac{\Gamma _{\kappa }(\alpha +\kappa)(n+1)^{\frac{\alpha }{ \kappa }}}{3\big(\eta (\zeta, \varepsilon)\big)^{\frac{\alpha }{\kappa }}} \Big[J_{\big(\varepsilon +\frac{n}{n+1}\eta (\zeta, \varepsilon)\big) ^{+}}^{\alpha, \kappa }\varphi \big(\varepsilon +\eta (\zeta, \varepsilon) \big)+J_{\big(\varepsilon +\frac{1}{n+1}\eta (\zeta, \varepsilon)\big) ^{+}}^{\alpha, \kappa }\varphi (\varepsilon)\Big]\Bigg]. \end{eqnarray*}$

Proof. Integration by parts, one can have

$\begin{eqnarray*} I_{1}&& = \int\limits_{0}^{1}\Big(\frac{2(1-\theta)^{\frac{\alpha}{\kappa} }-\theta^{\frac{\alpha}{\kappa}}}{3}\Big) \varphi^{\prime}\Big(\varepsilon+ \frac{1-\theta}{n+1}\eta(\zeta, \varepsilon)\Big)d\theta \notag \\ && = \frac{n+1}{3\eta(\zeta, \varepsilon)}\varphi(\varepsilon)+\frac{2(n+1)}{ 3\eta(\zeta, \varepsilon)}\varphi\Big(\varepsilon+\frac{1}{n+1} \eta(\zeta, \varepsilon)\Big) \notag \\ &&\quad-\frac{\alpha(n+1)^{\frac{\alpha}{\kappa}+1}}{3\kappa\big( \eta(\zeta, \varepsilon)\big)^{\frac{\alpha}{\kappa}+1}} \int\limits_{ \varepsilon}^{\varepsilon+\frac{1}{n+1}\eta(\zeta, \varepsilon)} \Big( \varepsilon+\frac{1}{n+1}\eta(\zeta, \varepsilon)-x\Big)^{\frac{\alpha}{\kappa} -1}\varphi(x)dx \notag \\ &&\quad-\frac{2\alpha(n+1)^{\frac{\alpha}{\kappa}+1 }}{3\kappa\big( \eta(\zeta, \varepsilon)\big)^{\frac{\alpha}{\kappa}}} \int\limits_{ \varepsilon}^{\varepsilon+\frac{1}{n+1}\eta(\zeta, \varepsilon)}\big( x-\varepsilon\big)^{\frac{\alpha}{\kappa}-1}\varphi(x)dx \end{eqnarray*}$

and

$\begin{eqnarray*} I_{2}&& = \int\limits_{0}^{1}\Big(\frac{\theta^{\frac{\alpha}{\kappa} }-2(1-\theta)^{\frac{\alpha}{\kappa}}}{3}\Big)\Big] \varphi^{\prime}\Big( \varepsilon+\frac{n+\theta}{n+1}\eta(\zeta, \varepsilon)\Big)d\theta \\ && = \frac{n+1}{3\big(\eta(\zeta, \varepsilon)\big)}\varphi\big( \varepsilon+\eta(\zeta, \varepsilon)\big)+\frac{2(n+1)}{3\eta(\zeta, \varepsilon)} \varphi\big(\varepsilon+\frac{n}{n+1}\eta(\zeta, \varepsilon) \big) \notag \\ &&\quad-\frac{\alpha(n+1)^{\frac{\alpha}{\kappa}}+1} {3\kappa\big( \eta(\zeta, \varepsilon)\big)^{\frac{\alpha}{\kappa}+1}}\int\limits_{ \varepsilon+\frac{n}{n+1} \eta(\zeta, \varepsilon)}^{\varepsilon+\eta(\zeta, \varepsilon)}\Big(x-\big(\varepsilon+\frac{n}{n+1}\eta(\zeta, \varepsilon) \big)\Big)^{\frac{\alpha}{\kappa}-1}\varphi(x)dx \notag \\ &&\quad-\frac{\alpha(n+1)^{\frac{\alpha}{\kappa}}+1} {3\kappa\big( \eta(\zeta, \varepsilon)\big)^{\frac{\alpha}{\kappa}+1}}\int\limits_{ \varepsilon+\frac{n}{n+1} \eta(\zeta, \varepsilon)}^{\varepsilon+\eta(\zeta, \varepsilon)}\Big(\big(\varepsilon+\eta(\zeta, \varepsilon)\big)-x\Big)^{ \frac{\alpha}{\kappa}-1}\varphi(x)dx. \end{eqnarray*}$

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

$\begin{eqnarray*} I_{1}+I_{2}&& = \frac{1}{6}\Big[\varphi(\varepsilon)+\varphi(\varepsilon+\eta( \zeta, \varepsilon))+2\varphi\big(\varepsilon+\frac{1}{n+1} \eta(\zeta, \varepsilon)\big)+ 2\varphi\big(\varepsilon+\frac{n}{n+1} \eta(\zeta, \varepsilon)\big)\Big] \notag \\ &&\quad-\frac{\alpha(n+1)^{\frac{\alpha}{\kappa}}}{6\kappa\big( \eta(\zeta, \varepsilon)\big)^{\frac{\alpha}{\kappa}}} \Big[ \int\limits_{\varepsilon}^{\varepsilon+\frac{1}{n+1}\eta(\zeta, \varepsilon)} \big(\varepsilon+\frac{1}{n+1}\eta(\zeta, \varepsilon)\big)^{\frac{\alpha}{ \kappa}-1} \varphi(x)dx \\ &&\quad+\int\limits_{\varepsilon+\frac{n}{n+1}\eta(\zeta, \varepsilon)}^{ \varepsilon+\eta(\zeta, \varepsilon)}\Big(x-\big(\varepsilon+\frac{n}{n+1} \eta(\zeta, \varepsilon)\big)\Big)^{\frac{\alpha}{\kappa}-1}\varphi(x)dx\Big] \\ &&\quad- \frac{\alpha(n+1)^{\frac{\alpha}{\kappa}}}{3\kappa\big( \eta(\zeta, \varepsilon)\big)^{\frac{\alpha}{\kappa}}} \Big[ \int\limits_{\varepsilon}^{\varepsilon+\frac{n}{n+1}\eta(\zeta, \varepsilon)} \big(\varepsilon+\eta(\zeta, \varepsilon)-x\big)^{\frac{\alpha}{\kappa}-1} \varphi(x)dx+\int\limits_{\varepsilon}^{\varepsilon+\frac{1}{n+1} \eta(\zeta, \varepsilon)}(x-\varepsilon)^{\frac{\alpha}{\kappa}-1}\varphi(x)dx \Big]. \end{eqnarray*}$

Using the fact that

$\begin{eqnarray*} \frac{1}{\kappa\Gamma_{\kappa}(\alpha)}\int\limits_{\varepsilon}^{ \varepsilon+\frac{1}{n+1} \eta(\zeta, \varepsilon)}\big(x-\varepsilon\big)^{ \frac{\alpha}{\kappa}-1}\varphi(x)dx&& = J_{(\varepsilon+\frac{1}{n+1} \eta(\zeta, \varepsilon))^{-}}^{\alpha, \kappa}\varphi(\varepsilon), \\ \frac{1}{\kappa\Gamma_{\kappa}(\alpha)}\int\limits_{\varepsilon+\frac{n}{n+1} \eta(\zeta, \varepsilon)}^{\varepsilon+ \eta(\zeta, \varepsilon)}\big( \varepsilon+\eta(\zeta, \varepsilon)-x\big)^{\frac{\alpha}{\kappa} -1}\varphi(x)dx&& = J_{(\varepsilon+\frac{n}{n+1} \eta(\zeta, \varepsilon))^{+}}^{\alpha, \kappa}\varphi\big(\varepsilon+\eta(\zeta, \varepsilon)\big), \\ \frac{1}{\kappa\Gamma_{\kappa}(\alpha)}\int\limits_{\varepsilon}^{ \varepsilon+\frac{1}{n+1}\eta(\zeta, \varepsilon) }\big(\varepsilon+\frac{1}{ n+1}\eta(\zeta, \varepsilon)-x\big)^{\frac{\alpha}{\kappa}-1} \varphi(x)dx&& = J_{(\varepsilon)^{+}}^{\alpha, \kappa} \varphi\big(\varepsilon+ \frac{1}{n+1}\eta(\zeta, \varepsilon)\big), \\ \frac{1}{\kappa\Gamma_{\kappa}(\alpha)}\int\limits_{\varepsilon+\frac{n}{n+1} \eta(\zeta, \varepsilon)}^{\varepsilon+\eta(\zeta, \varepsilon) }\big(x-\big( \varepsilon+\frac{1}{n+1}\eta(\zeta, \varepsilon)\big))^{\frac{\alpha}{\kappa} -1}\varphi(x)dx&& = J_{(\varepsilon+\eta(\zeta, \varepsilon))^{-}}^{\alpha, \kappa} \varphi\big(\varepsilon+\frac{n}{n+1}\eta(\zeta, \varepsilon)\big), \\ \end{eqnarray*}$

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:

$\begin{eqnarray} &&|\Psi (\varepsilon, \zeta ;n, \alpha, \kappa)| \\ &\leq &\frac{\eta (\zeta, \varepsilon)}{2(n+1)}\Big(\int\limits_{0}^{1} \bigg\vert\frac{2(1-\theta)^{\frac{\alpha }{\kappa }}-\theta ^{\frac{\alpha }{\kappa }}}{3}\bigg\vert^{\mu }d\theta \Big)^{\frac{1}{\mu }} \\ &&\quad \times \Bigg[\Big[\Big(1-\frac{1}{(n+1)^{s}(s+1)}|\varphi ^{\prime }(\varepsilon)|^{\lambda }+\frac{1}{(n+1)^{s}(s+1)}|\varphi ^{\prime }(\zeta)|^{\lambda }\Big)\Big]^{\frac{1}{\lambda }} \\ &&\quad +\Big[\frac{1}{(n+1)^{s}(s+1)}|\varphi ^{\prime }(\varepsilon )|^{\lambda }+\Big(2^{1-s}-\frac{1}{(n+1)^{s}(s+1)}\Big)|\varphi ^{\prime }(\zeta)|^{\lambda }\Big]^{\frac{1}{\lambda }}\Bigg]. \end{eqnarray}$

(2.2)

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

$\begin{eqnarray} &&|\Psi (\varepsilon, \zeta ;n, \alpha, \kappa)|\leq \frac{\eta (\zeta , \varepsilon)}{2(n+1)}\Big[\int\limits_{0}^{1}\Big\vert\Big(\frac{ 2(1-\theta)^{\frac{\alpha }{\kappa }}-\theta ^{\frac{\alpha }{\kappa }}}{3} \Big)\Big\vert\Big\vert\varphi ^{\prime }\Big(\varepsilon +\frac{1-\theta }{ n+1}\eta (\zeta, \varepsilon)\Big)\Big\vert d\theta \\ &&+\Big\vert\int\limits_{0}^{1}\Big(\frac{\theta ^{\frac{\alpha }{\kappa } }-2(1-\theta)^{\frac{\alpha }{\kappa }}}{3}\Big)\Big\vert\Big\vert\varphi ^{\prime }\Big(\varepsilon +\frac{n+\theta }{n+1}\eta (\zeta, \varepsilon) \Big)\Big\vert d\theta \Big] \\ &\leq &\frac{\eta (\zeta, \varepsilon)}{2(n+1)}\Bigg[\Big( \int\limits_{0}^{1}\Big\vert\Big(\frac{2(1-\theta)^{\frac{\alpha }{\kappa } }-\theta ^{\frac{\alpha }{\kappa }}}{3}\Big)\Big\vert^{\mu }d\theta \Big)^{ \frac{1}{\mu }} \\ &&\quad \times \Big[\int\limits_{0}^{1}\Bigg(\Big(1-\Big(\frac{1-\theta }{n+1 }\Big)^{s}\Big)|\varphi ^{\prime }(\varepsilon)|^{\lambda }+\Big(\frac{ 1-\theta }{n+1}\Big)^{s}|\varphi ^{\prime }(\zeta)|^{\lambda }\Bigg)d\theta \Big]^{\frac{1}{\lambda }} \\ &&\quad +\Big(\int\limits_{0}^{1}\Big\vert\Big(\frac{\theta ^{\frac{\alpha }{ \kappa }}-2(1-\theta)^{\frac{\alpha }{\kappa }}}{3}\Big)\Big\vert^{\mu }d\theta \Big)^{\frac{1}{\mu }}\Big[\int\limits_{0}^{1}\Bigg(\Big(1-\Big( \frac{n+\theta }{n+1}\Big)^{s}\Big)|\varphi ^{\prime }(\varepsilon )|^{\lambda }+\Big(\frac{n+\theta }{n+1}\Big)^{s}|\varphi ^{\prime }(\zeta )|^{\lambda }\Bigg)d\theta \Big]^{\frac{1}{\lambda }}\Bigg]. \end{eqnarray}$

(2.3)

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

$\begin{eqnarray} &&\int\limits_{0}^{1}\Bigg(1-\Big(\frac{n+\theta }{n+1}\Big)^{s}|\varphi ^{\prime }(\varepsilon)|^{\lambda }+\Big(\frac{n+\theta }{n+1}\Big)|\varphi ^{\prime }(\zeta)|^{\lambda }\Bigg)d\theta \\ &\leq &\int\limits_{0}^{1}\Big[\Big(\frac{1-\theta }{n+1}\Big)|\varphi ^{\prime }(\varepsilon)|^{\lambda }+\Big(2^{1-s}-\Big(\frac{1-\theta }{n+1} \Big)^{s}\Big)|\varphi ^{\prime }(\zeta)|^{\lambda }\Big]d\theta \\ & = &\frac{1}{(n+1)^{s}(s+1)}|\varphi ^{\prime }(\varepsilon)|^{\lambda }+ \Big(2^{1-s}-\frac{1}{(n+1)^{s}(s+1)}\Big)|\varphi ^{\prime }(\zeta )|^{\lambda } \end{eqnarray}$

(2.4)

and

$\begin{eqnarray} &&\int\limits_{0}^{1}\Bigg[\Bigg(1-\Big(\frac{1-\theta }{n+1}\Big)^{s}\Bigg) |\varphi ^{\prime }(\varepsilon)|^{\lambda }+\Big(\frac{1-\theta }{n+1}\Big) ^{s}|\varphi ^{\prime }(\zeta)|^{\lambda }\Bigg]d\theta \\ & = &\Big(1-\frac{1}{(n+1)^{s}(s+1)}|\varphi ^{\prime }(\varepsilon )|^{\lambda }+\frac{1}{(n+1)^{s}(s+1)}|\varphi ^{\prime }(\zeta)|^{\lambda } \Big). \end{eqnarray}$

(2.5)

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

$\begin{eqnarray*} &&\Big\vert \frac{1}{6}\Big[\varphi(\varepsilon)+4\varphi\big(\frac{ 2\varepsilon+ \eta(\zeta, \varepsilon)}{2}\big)+\varphi(\varepsilon+\eta( \zeta, \varepsilon))\Big]-\frac{1}{\eta(\zeta, \varepsilon)} \int\limits_{\varepsilon}^{\varepsilon+\eta(\zeta, \varepsilon)} \varphi(x)dx \Big\vert \\ &&\leq\frac{\eta(\zeta, \varepsilon)}{4}\Big[\frac{1+2^{\mu+1}}{ 3^{\mu+1}(\mu+1)}\Big]^{\frac{1}{\mu}}\Big[\Big(\frac{3}{4}\Big)^{\frac{1}{ \lambda}}+ \Big(\frac{1}{4}\Big)^{\frac{1}{\lambda}}\Big]\Big[\vert \varphi^{\prime}(\varepsilon)\vert+\vert \varphi^{\prime}(\zeta)\vert\Big]. \end{eqnarray*}$

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

$\begin{eqnarray*} &&\Big\vert \frac{1}{6}\Big[\varphi(\varepsilon)+4\varphi\big(\frac{ \varepsilon+\zeta}{2}\big)+\varphi(\zeta)\Big]-\frac{1}{\zeta-\varepsilon} \int\limits_{\varepsilon}^{\zeta} \varphi(x)dx \Big\vert \\ &&\leq\frac{\zeta-\varepsilon}{4}\Big[\frac{1+2^{\mu+1}}{3^{\mu+1}(\mu+1)} \Big]^{\frac{1}{\mu}}\Big[\Big(\frac{3}{4}\Big)^{\frac{1}{\lambda}}+ \Big( \frac{1}{4}\Big)^{\frac{1}{\lambda}}\Big]\Big[\vert \varphi^{\prime}(\varepsilon)\vert+\vert \varphi^{\prime}(\zeta)\vert\Big]. \end{eqnarray*}$

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

$\begin{equation*} \sum\limits_{i = 1}^{n}(\omega_{i}+\nu_{i})^{j}\leq\sum\limits_{i = 1}^{n} (\omega_{i})^{j}+\sum\limits_{i = 1}^{n} (\nu_{i})^{j} \end{equation*}$

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:

$\begin{eqnarray} &&|\Psi (\varepsilon, \zeta ;n, \alpha, \kappa)| \\ &\leq &\frac{\eta (\zeta, \varepsilon)}{2(n+1)}\Big\{(\Delta _{1}+\Delta _{3})|\varphi ^{\prime }(\varepsilon)|+(\Delta _{1}+\Delta _{2})|\varphi ^{\prime }(\zeta)|\Big\}, \end{eqnarray}$

(2.6)

where

$\begin{equation*} \Delta _{1} = \frac{2-4\Big(1-\frac{2^{\frac{\kappa }{\alpha }}}{2^{\frac{ \kappa }{\alpha }}+1}\Big)^{\frac{\alpha }{\kappa }+s+1}}{3(n+1)^{s}(\frac{ \alpha }{\kappa }+s+1)}+\frac{1}{3(n+1)^{s}}\Big[\beta (\frac{\alpha }{ \kappa }+1, s+1)-2\beta \big(\frac{2^{\frac{\kappa }{\alpha }}}{2^{\frac{ \kappa }{\alpha }}+1};\frac{\alpha }{\kappa }+1, s+1\big)\Big], \end{equation*}$

$\begin{eqnarray*} \Delta _{2} & = &\frac{2^{1-s}\big\{3-2\Big(\frac{2^{\frac{\kappa }{\alpha }}}{ 2^{\frac{\alpha }{\kappa }}+1}\Big)^{\frac{\alpha }{\kappa }+1}-4\Big(1- \frac{2^{\frac{\kappa }{\alpha }}}{2^{\frac{\alpha }{\kappa }}+1}\Big)^{ \frac{\alpha }{\kappa }+1}\big\}}{3(\frac{\alpha }{\kappa }+1)}-\frac{2-4 \Big(1-\frac{2^{\frac{\kappa }{\alpha }}}{2^{\frac{\kappa }{\alpha }}+1}\Big) ^{\frac{\alpha }{\kappa }+s+1}}{3(n+1)^{s}(\frac{\alpha }{\kappa }+s+1)} \\ &&\quad +\frac{1}{3(n+1)^{s}}\Big[2\beta \big(\frac{2^{\frac{\kappa }{\alpha }}}{2^{\frac{\alpha }{\kappa }}+1};\frac{\alpha }{\kappa }+1, s+1\big)-\beta \big(\frac{\alpha }{\kappa }+1, s+1\big)\Big] \end{eqnarray*}$

and

$\begin{eqnarray*} \Delta _{3} & = &\frac{\big\{3-2\Big(\frac{2^{\frac{\kappa }{\alpha }}}{2^{ \frac{\alpha }{\kappa }}+1}\Big)^{\frac{\alpha }{\kappa }+1}-4\Big(1-\frac{ 2^{\frac{\kappa }{\alpha }}}{2^{\frac{\alpha }{\kappa }}+1}\Big)^{\frac{ \alpha }{\kappa }+1}\big\}}{3(\frac{\alpha }{\kappa }+1)}-\frac{2-4\Big(1- \frac{2^{\frac{\kappa }{\alpha }}}{2^{\frac{\kappa }{\alpha }}+1}\Big)^{ \frac{\alpha }{\kappa }+s+1}}{3(n+1)^{s}(\frac{\alpha }{\kappa }+s+1)} \\ &&\quad +\frac{1}{3(n+1)^{s}}\Big[2\beta \big(\frac{2^{\frac{\kappa }{\alpha }}}{2^{\frac{\alpha }{\kappa }}+1};\frac{\alpha }{\kappa }+1, s+1\big)-\beta \big(\frac{\alpha }{\kappa }+1, s+1\big)\Big]. \end{eqnarray*}$

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

$\begin{eqnarray} &&|\Psi (\varepsilon, \zeta ;n, \alpha, \kappa)| \\ &\leq &\frac{\eta (\zeta, \varepsilon)}{2(n+1)}\Bigg[\int\limits_{0}^{1} \bigg\vert\frac{2(1-\theta)^{\frac{\alpha }{\kappa }}-\theta ^{\frac{\alpha }{\kappa }}}{3}\bigg\vert\Big[\bigg(1-\Big(\frac{1-\theta }{n+1}\Big)^{s} \bigg)|\varphi ^{\prime }(\varepsilon)|+\Big(\frac{1-\theta }{n+1}\Big) ^{s}|\varphi ^{\prime }(\zeta)|\Big]d\theta \\ &&\quad +\int\limits_{0}^{1}\bigg\vert\frac{\theta ^{\frac{\alpha }{\kappa } }-2(1-\theta)^{\frac{\alpha }{\kappa }}}{3}\bigg\vert\Big[\bigg(1-\Big( \frac{n+\theta }{n+1}\Big)^{s}\bigg)|\varphi ^{\prime }(\varepsilon)|+\Big( \frac{n+\theta }{n+1}\Big)^{s}|\varphi ^{\prime }(\zeta)|\Big]d\theta \Bigg] \end{eqnarray}$

(2.7)

From (2.7), we have

$\begin{eqnarray} &&\int\limits_{0}^{1}\bigg\vert\frac{\theta ^{\frac{\alpha }{\kappa } }-2(1-\theta)^{\frac{\alpha }{\kappa }}}{3}\bigg\vert\Big[\bigg(1-\Big( \frac{n+\theta }{n+1}\Big)^{s}\bigg)|\varphi ^{\prime }(\varepsilon)|+\Big( \frac{n+\theta }{n+1}\Big)^{s}|\varphi ^{\prime }(\zeta)|\Big]d\theta \\ &\leq &|\varphi ^{\prime }(\varepsilon)|\int\limits_{0}^{1}\bigg\vert\frac{ \theta ^{\frac{\alpha }{\kappa }}-2(1-\theta)^{\frac{\alpha }{\kappa }}}{3} \bigg\vert\Big(\frac{1-\theta }{n+1}\Big)^{s}d\theta +|\varphi ^{\prime }(\zeta)|\int\limits_{0}^{1}\bigg\vert\frac{\theta ^{\frac{\alpha }{\kappa } }-2(1-\theta)^{\frac{\alpha }{\kappa }}}{3}\bigg\vert\Big(2^{1-s}-\frac{ 1-\theta }{n+1}\Big)^{s}d\theta, \end{eqnarray}$

(2.8)

By taking into account

$\begin{eqnarray} \Delta _{1} & = &\int\limits_{0}^{1}\bigg\vert\frac{\theta ^{\frac{\alpha }{ \kappa }}-2(1-\theta)^{\frac{\alpha }{\kappa }}}{3}\bigg\vert\Big(\frac{ 1-\theta }{n+1}\Big)^{s}d\theta \\ & = &\frac{2-4\Big(1-\frac{2^{\frac{\kappa }{\alpha }}}{2^{\frac{\kappa }{ \alpha }}+1}\Big)^{\frac{\alpha }{\kappa }+s+1}}{3(n+1)^{s}(\frac{\alpha }{ \kappa }+s+1)}+\frac{1}{3(n+1)^{s}}\Big[\beta (\frac{\alpha }{\kappa } +1, s+1)-2\beta \big(\frac{2^{\frac{\kappa }{\alpha }}}{2^{\frac{\kappa }{ \alpha }}+1};\frac{\alpha }{\kappa }+1, s+1\big)\Big], \end{eqnarray}$

(2.9)

$\begin{eqnarray} \Delta _{2} & = &\int\limits_{0}^{1}\bigg\vert\frac{\theta ^{\frac{\alpha }{ \kappa }}-2(1-\theta)^{\frac{\alpha }{\kappa }}}{3}\bigg\vert\Big(2^{1-s}- \frac{1-\theta }{n+1}\Big)^{s}d\theta \\ & = &\frac{2^{1-s}\big\{3-2\Big(\frac{2^{\frac{\kappa }{\alpha }}}{2^{\frac{ \alpha }{\kappa }}+1}\Big)^{\frac{\alpha }{\kappa }+1}-4\Big(1-\frac{2^{ \frac{\kappa }{\alpha }}}{2^{\frac{\alpha }{\kappa }}+1}\Big)^{\frac{\alpha }{\kappa }+1}\big\}}{3(\frac{\alpha }{\kappa }+1)}-\frac{2-4\Big(1-\frac{2^{ \frac{\kappa }{\alpha }}}{2^{\frac{\kappa }{\alpha }}+1}\Big)^{\frac{\alpha }{\kappa }+s+1}}{3(n+1)^{s}(\frac{\alpha }{\kappa }+s+1)} \\ &&\quad +\frac{1}{3(n+1)^{s}}\Big[2\beta \big(\frac{2^{\frac{\kappa }{\alpha }}}{2^{\frac{\alpha }{\kappa }}+1};\frac{\alpha }{\kappa }+1, s+1\big)-\beta \big(\frac{\alpha }{\kappa }+1, s+1\big)\Big] \end{eqnarray}$

(2.10)

and

$\begin{eqnarray} \Delta _{3} & = &\int\limits_{0}^{1}\bigg\vert\frac{\theta ^{\frac{\alpha }{ \kappa }}-2(1-\theta)^{\frac{\alpha }{\kappa }}}{3}\bigg\vert\Big(1-\frac{ 1-\theta }{n+1}\Big)^{s}d\theta \\ & = &\frac{\big\{3-2\Big(\frac{2^{\frac{\kappa }{\alpha }}}{2^{\frac{\alpha }{ \kappa }}+1}\Big)^{\frac{\alpha }{\kappa }+1}-4\Big(1-\frac{2^{\frac{\kappa }{\alpha }}}{2^{\frac{\alpha }{\kappa }}+1}\Big)^{\frac{\alpha }{\kappa }+1} \big\}}{3(\frac{\alpha }{\kappa }+1)}-\frac{2-4\Big(1-\frac{2^{\frac{\kappa }{\alpha }}}{2^{\frac{\kappa }{\alpha }}+1}\Big)^{\frac{\alpha }{\kappa } +s+1}}{3(n+1)^{s}(\frac{\alpha }{\kappa }+s+1)} \\ &&\quad +\frac{1}{3(n+1)^{s}}\Big[2\beta \big(\frac{2^{\frac{\kappa }{\alpha }}}{2^{\frac{\alpha }{\kappa }}+1};\frac{\alpha }{\kappa }+1, s+1\big)-\beta \big(\frac{\alpha }{\kappa }+1, s+1\big)\Big], \end{eqnarray}$

(2.11)

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:

$\begin{eqnarray} &&|\Psi (\varepsilon, \zeta ;n, \alpha, \kappa)| \\ &\leq &\frac{\eta (\zeta, \varepsilon)}{2(n+1)}(\Delta _{0})^{1-\frac{1}{ \lambda }}\bigg\{\Big(\Delta _{1}|\varphi ^{\prime }(\varepsilon)|^{\lambda }+\Delta _{3}|\varphi ^{\prime }(\zeta)|^{\lambda }\Big)^{\frac{1}{\lambda } }+\Big(\Delta _{1}|\varphi ^{\prime }(\varepsilon)|^{\lambda }+\Delta _{2}|\varphi ^{\prime }(\zeta)|^{\lambda }\Big)^{\frac{1}{\lambda }}\bigg\}, \end{eqnarray}$

(2.12)

where

$\begin{equation*} \Delta _{0} = \frac{\big\{3-2\Big(\frac{2^{\frac{\kappa }{\alpha }}}{2^{\frac{ \alpha }{\kappa }}+1}\Big)^{\frac{\alpha }{\kappa }+1}-4\Big(1-\frac{2^{ \frac{\kappa }{\alpha }}}{2^{\frac{\alpha }{\kappa }}+1}\Big)^{\frac{\alpha }{\kappa }+1}\big\}}{3(\frac{\alpha }{\kappa }+1)}. \end{equation*}$

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

$\begin{eqnarray*} &&|\Psi (\varepsilon, \zeta ;n, \alpha, \kappa)| \\ &\leq &\frac{\eta (\zeta, \varepsilon)}{2(n+1)}\Bigg[\Bigg( \int\limits_{0}^{1}\bigg\vert\frac{2(1-\theta)^{\frac{\alpha }{\kappa } }-\theta ^{\frac{\alpha }{\kappa }}}{3}\bigg\vert d\theta \Bigg)^{1-\frac{1}{ \lambda }} \\ &&\quad \times \Bigg[\int\limits_{0}^{1}\bigg\vert\frac{2(1-\theta)^{\frac{ \alpha }{\kappa }}-\theta ^{\frac{\alpha }{\kappa }}}{3}\bigg\vert\bigg\vert \varphi ^{\prime }\Big(\varepsilon +\frac{1-\theta }{n+1}\eta (\zeta , \varepsilon)\Big)\bigg\vert^{\lambda }d\theta \bigg)^{\frac{1}{\lambda }} \\ &&\quad +\int\limits_{0}^{1}\bigg\vert\frac{\theta ^{\frac{\alpha }{\kappa } }-2(1-\theta)^{\frac{\alpha }{\kappa }}}{3}\bigg\vert\bigg\vert\varphi ^{\prime }\Big(\varepsilon +\frac{n+\theta }{n+1}\eta (\zeta, \varepsilon) \Big)\bigg\vert^{\lambda }d\theta \bigg)^{\frac{1}{\lambda }}\Bigg] \end{eqnarray*}$

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

$\begin{eqnarray} &&|\Psi (\varepsilon, \zeta ;n, \alpha, \kappa)| \\ &\leq &\frac{\eta (\zeta, \varepsilon)}{2(n+1)}\Bigg[\Bigg( \int\limits_{0}^{1}\bigg\vert\frac{2(1-\theta)^{\frac{\alpha }{\kappa } }-\theta ^{\frac{\alpha }{\kappa }}}{3}\bigg\vert d\theta \Bigg)^{1-\frac{1}{ \lambda }} \\ &&\times \Bigg[\int\limits_{0}^{1}\bigg\vert\frac{2(1-\theta)^{\frac{\alpha }{\kappa }}-\theta ^{\frac{\alpha }{\kappa }}}{3}\bigg\vert\bigg[\Bigg(\bigg( 1-\Big(\frac{1-\theta }{n+1}\Big)^{s}\bigg)|\varphi ^{\prime }(\varepsilon )|^{\lambda }+\Big(\frac{1-\theta }{n+1}\Big)^{\lambda }|\varphi ^{\prime }(\zeta)|^{\lambda }\Bigg)d\theta \bigg]^{\frac{1}{\lambda }} \\ &&\quad +\int\limits_{0}^{1}\bigg\vert\frac{\theta ^{\frac{\alpha }{\kappa } }-2(1-\theta)^{\frac{\alpha }{\kappa }}}{3}\bigg\vert\bigg[\Bigg(\bigg(1- \Big(\frac{n+\theta }{n+1}\Big)^{s}\bigg)|\varphi ^{\prime }(\varepsilon )|^{\lambda }+\Big(\frac{n+\theta }{n+1}\Big)^{\lambda }|\varphi ^{\prime }(\zeta)|^{\lambda }\Bigg)d\theta \bigg]^{\frac{1}{\lambda }}, \Bigg] \end{eqnarray}$

(2.13)

using the fact that

$\begin{eqnarray} \Delta _{0} & = &\int\limits_{0}^{1}\bigg\vert\frac{2(1-\theta)^{\frac{\alpha }{\kappa }}-\theta ^{\frac{\alpha }{\kappa }}}{3}\bigg\vert d\theta \\ & = &\frac{\big\{3-2\Big(\frac{2^{\frac{\kappa }{\alpha }}}{2^{\frac{\alpha }{ \kappa }}+1}\Big)^{\frac{\alpha }{\kappa }+1}-4\Big(1-\frac{2^{\frac{\kappa }{\alpha }}}{2^{\frac{\alpha }{\kappa }}+1}\Big)^{\frac{\alpha }{\kappa }+1} \big\}}{3(\frac{\alpha }{\kappa }+1)}. \end{eqnarray}$

(2.14)

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

$\begin{eqnarray*} &&\Big\vert \frac{1}{6}\Big(\varphi(\varepsilon)+4\varphi\big(\frac{ 2\varepsilon+\eta(\zeta, \varepsilon)}{2}\big)+\varphi(\varepsilon+\eta( \zeta, \varepsilon))\Big) -\frac{1}{\eta(\zeta, \varepsilon)} \int\limits_{\varepsilon}^{\varepsilon+\eta(\zeta, \varepsilon)}\varphi(x)dx \Big\vert \\ &&\leq\frac{\eta(\zeta, \varepsilon)}{4}\big(\frac{5}{18}\big)^{1-\frac{1}{ \lambda}}\Big[\big(\frac{61}{324}\big)^{\frac{1}{\lambda}}+\big(\frac{29}{324 }\big)^{\frac{1}{\lambda}}\Big] \lbrack \vert \varphi^{\prime}(\varepsilon)\vert+ \vert \varphi^{\prime}(\zeta)\vert]. \end{eqnarray*}$

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

$\begin{eqnarray*} &&\Big\vert \frac{1}{6}\Big(\varphi(\varepsilon)+4\varphi\big(\frac{ \varepsilon+\zeta}{2}\big)+\varphi(\zeta)\Big) -\frac{1}{\zeta-\varepsilon} \int\limits_{\varepsilon}^{\zeta}\varphi(x)dx\Big\vert \\ &&\leq\frac{\zeta-\varepsilon}{4}\big(\frac{5}{18}\big)^{1-\frac{1}{\lambda}} \Big[\big(\frac{61}{324}\big)^{\frac{1}{\lambda}}+\big(\frac{29}{324}\big)^{ \frac{1}{\lambda}}\Big] \lbrack \vert \varphi^{\prime}(\varepsilon)\vert+ \vert \varphi^{\prime}(\zeta)\vert]. \end{eqnarray*}$

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:

$\begin{equation} |\Psi (\varepsilon, \zeta ;n, \alpha, \kappa)|\leq \frac{(\mathcal{M}-m)\eta (\zeta, \varepsilon)}{2(n+1)}\Delta _{0}, \end{equation}$

(2.15)

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

Proof. From Lemma 2.1, we have

$\begin{eqnarray} \Psi (\varepsilon, \zeta ;n, \alpha, \kappa)| & = &\frac{\eta (\zeta , \varepsilon)}{2(n+1)}\bigg[\int\limits_{0}^{1}\bigg(\frac{2(1-\theta)^{ \frac{\alpha }{\kappa }}-\theta ^{\frac{\alpha }{\kappa }}}{3}\bigg)\Bigg( \varphi ^{\prime }\bigg(\varepsilon +\frac{1-\theta }{n+1}\eta (\zeta , \varepsilon)\bigg)-\frac{m+\mathcal{M}}{2}\Bigg)d\theta \\ &&\quad +\int\limits_{0}^{1}\bigg(\frac{\theta ^{\frac{\alpha }{\kappa } }-2(1-\theta)^{\frac{\alpha }{\kappa }}}{3}\bigg)\Bigg(\varphi ^{\prime } \bigg(\varepsilon +\frac{n+\theta }{n+1}\eta (\zeta, \varepsilon)\bigg)- \frac{m+\mathcal{M}}{2}\Bigg)d\theta \bigg] \end{eqnarray}$

(2.16)

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

$\begin{equation*} \Big\vert\varphi ^{\prime }\Big(\varepsilon +\frac{1-\theta }{n+1}\eta (\zeta, \varepsilon)\Big)-\frac{m+\mathcal{M}}{2}\Big\vert\leq \frac{ \mathcal{M}-m}{2}, \end{equation*}$

similarly,

$\begin{equation*} \Big\vert\varphi ^{\prime }\Big(\varepsilon +\frac{n+\theta }{n+1}\eta (\zeta, \varepsilon)\Big)-\frac{m+\mathcal{M}}{2}\Big\vert\leq \frac{ \mathcal{M}-m}{2}, \end{equation*}$

Inequality (2.16) implies that

$\begin{eqnarray*} |\Psi (\varepsilon, \zeta ;n, \alpha, \kappa)| &\leq &\frac{(\mathcal{M} -m)\eta (\zeta, \varepsilon)}{2(n+1)}\int\limits_{0}^{1}\Big\vert\frac{ 2(1-\theta)^{\frac{\alpha }{\kappa }}-\theta ^{\frac{\alpha }{\kappa }}}{3} \Big\vert d\theta \\ & = &\frac{(\mathcal{M}-m)\eta (\zeta, \varepsilon)}{2(n+1)}\Delta _{0}. \end{eqnarray*}$

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

$\begin{equation*} |\Psi (\varepsilon, \zeta ;n, \alpha, \kappa)|\leq \frac{7(\mathcal{M} -m)(\zeta -\varepsilon)}{36(n+1)}. \end{equation*}$

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:

$\begin{eqnarray} &&|\Psi (\varepsilon, \zeta ;n, \alpha, \kappa)| \\ &\leq &\frac{\mathfrak{L}\big(\eta (\zeta, \varepsilon)\big)^{2}}{2(n+1)^{2} }\Big[(n-1)\Delta _{0}-\frac{4\Big(\frac{2^{\frac{\alpha }{\kappa }}}{2^{ \frac{\alpha }{\kappa }}+1}\Big)^{\frac{\alpha }{\kappa }+2}-2}{3(\frac{ \alpha }{\kappa })+2}+\frac{8}{3}\beta \Big(\frac{2^{\frac{\alpha }{\kappa }} }{2^{\frac{\alpha }{\kappa }}+1};\frac{\alpha }{\kappa }+1, 2\Big)-\frac{4}{3} \beta \Big(\frac{\alpha }{\kappa }+1, 2\Big)\Big], \end{eqnarray}$

(2.17)

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

Proof. From Lemma 2.1, we have

$\begin{eqnarray*} &&\Psi(\varepsilon, \zeta;n, \alpha, \kappa)\vert \\ && = \frac{\eta(\zeta, \varepsilon)}{2(n+1)}\bigg[\int\limits_{0}^{1}\bigg( \frac{2(1-\theta)^{\frac{\alpha}{\kappa}}-\theta^{ \frac{\alpha}{\kappa}}}{3} \bigg)\Bigg[\varphi^{\prime}\bigg(\varepsilon+\frac{1-\theta}{n+1} \eta(\zeta, \varepsilon)\bigg)-\varphi^{\prime}\bigg(\varepsilon+ \frac{ n+\theta}{n+1}\eta(\zeta, \varepsilon)\bigg)\bigg]d\theta. \end{eqnarray*}$

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

$\begin{eqnarray*} \Big\vert\varphi^{\prime}\bigg(\varepsilon+\frac{1-\theta}{n+1} \eta(\zeta, \varepsilon)\bigg)-\varphi^{\prime}\bigg(\varepsilon+ \frac{ n+\theta}{n+1}\eta(\zeta, \varepsilon)\bigg)\Big\vert\leq\mathfrak{L} \eta(\zeta, \varepsilon)\Big(\frac{n-1+2t}{n+1}\Big) \end{eqnarray*}$

Thus

$\begin{eqnarray*} &&\vert\Psi(\varepsilon, \zeta;n, \alpha, \kappa)\vert \\ &&\leq\frac{\eta(\zeta, \varepsilon)}{2(n+1)}\bigg[\int\limits_{0}^{1} \bigg\vert\frac{2(1-\theta)^{\frac{\alpha}{\kappa}}-\theta^{ \frac{\alpha}{ \kappa}}}{3}\bigg\vert\bigg\vert\varphi^{\prime}\bigg(\varepsilon+\frac{ 1-\theta}{n+1}\eta(\zeta, \varepsilon)\bigg)-\varphi^{\prime}\bigg( \varepsilon+ \frac{n+\theta}{n+1}\eta(\zeta, \varepsilon)\bigg\vert\bigg] d\theta \\ &&\leq \frac{\mathfrak{L}\big(\eta(\zeta, \varepsilon)\big)^{2}}{2(n+1)} \int\limits_{0}^{1}\bigg\vert\frac{2(1-\theta)^{\frac{\alpha}{\kappa} }-\theta^{ \frac{\alpha}{\kappa}}}{3}\bigg\vert\Big(\frac{n-1+2t}{n+1}\Big) d\theta \\ && = \frac{\mathfrak{L}\big(\eta(\zeta, \varepsilon)\big)^{2}}{2(n+1)^{2}}\Big[ (n-1)\Delta_{0}-\frac{4\Big(\frac{2^{\frac{\alpha}{\kappa}}}{ 2^{\frac{\alpha }{\kappa}}+1}\Big)^{\frac{\alpha}{\kappa}+2}-2}{3(\frac{\alpha}{\kappa})+2}+ \frac{8}{3}\beta\Big(\frac{2^{\frac{\alpha}{\kappa}}}{ 2^{\frac{\alpha}{ \kappa}}+1};\frac{\alpha}{\kappa}+1, 2\Big)-\frac{4}{3}\beta\Big(\frac{\alpha }{\kappa}+1, 2\Big)\Big]. \end{eqnarray*}$

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

$\begin{eqnarray*} \vert\Psi(\varepsilon, \zeta;n, \alpha, \kappa)\vert\leq\frac{\mathfrak{L} \big( \zeta-\varepsilon\big)^{2}}{2(n+1)^{2}}\Big[\frac{7n-11}{18}+\frac{22}{135}+ \frac{8}{3}\beta\Big(\frac{2}{3};2, 2\Big)\Big]. \end{eqnarray*}$

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

$\begin{eqnarray} \mathcal{D}_{\mathcal{F}}(z, w): = \int\limits_{\psi}z(x)\mathcal{F}\Big(\frac{ z(x)}{w(x)}\Big)d\varrho(x), \quad z, w\in \Lambda. \end{eqnarray}$

(3.1)

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

Consider the following Hermite-Hadamard divergence

$\begin{eqnarray} \mathcal{D}^{\mathcal{F}}_{HH}(z, w): = \int\limits_{\psi}\frac{ \int\limits_{1}^{\frac{w(x)}{z(x)}}\mathcal{F}(t)dt}{\frac{w(x)}{z(x)}-1} d\varrho(x), \quad w, z\in\Lambda, \end{eqnarray}$

(3.2)

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:

$\begin{eqnarray} &&\Big\vert\frac{1}{6}\Big[\mathcal{D}_{\mathcal{F}}(w, z)+4\int\limits_{\psi }z(x)\mathcal{F}\Big(\frac{z(x)+w(x)}{2w(x)}d\varrho (x)\Big)\Big]-\mathcal{D }_{HH}^{\mathcal{F}}(w, z)\Big\vert \\ &\leq &\frac{(1+2^{\lambda +1})^{\frac{1}{\lambda }}}{4\big(3^{\lambda +1}(1+\lambda)\big)^{\frac{1}{\lambda }}}\bigg[\Big(\frac{3}{4}\Big)^{\frac{ 1}{\lambda }}+\Big(\frac{1}{4}\Big)^{\frac{1}{\lambda }}\bigg]\times \\ &&\Bigg[|\mathcal{F}^{\prime }(1)|\int\limits_{\psi }|z(x)-w(x)|d\varrho (x)+\int\limits_{\psi }|z(x)-w(x)|\big\vert\mathcal{F}^{\prime }\frac{z(x)}{ w(x)}\big\vert d\varrho (x)\Bigg]. \end{eqnarray}$

(3.3)

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

$\begin{eqnarray} &&\Big\vert\frac{1}{6}\bigg[4\int\limits_{\psi _{2}}w(x)\mathcal{F}\Big( \frac{w(x)+z(x)}{2w(x)}\Big)d\varrho (x)+\int\limits_{\psi _{2}}w(x)\mathcal{ F}\Big(\frac{z(x)}{w(x)}\Big)d\varrho (x)\bigg]-\int\limits_{\psi _{2}}w(x) \frac{\int\limits_{1}^{\frac{z(x)}{w(x)}}\mathcal{F}(t)dt}{\frac{z(x)}{w(x)} -1}d\varrho (x)\Big\vert \\ &\leq &\frac{(1+2^{\lambda +1})^{\frac{1}{\lambda }}}{4\big(3^{\lambda +1}(1+\lambda)\big)^{\frac{1}{\lambda }}}\bigg[\Big(\frac{3}{4}\Big)^{\frac{ 1}{\lambda }}+\Big(\frac{1}{4}\Big)^{\frac{1}{\lambda }}\bigg]\times \\ &&\Bigg[|\mathcal{F}^{\prime }(1)|\int\limits_{\psi _{2}}|w(x)-z(x)|d\varrho (x)+\int\limits_{\psi _{2}}|w(x)-z(x)|\big\vert\mathcal{F}^{\prime }\frac{ z(x)}{w(x)}\big\vert d\varrho (x)\Bigg]. \end{eqnarray}$

(3.4)

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

$\begin{eqnarray} &&\Big\vert\frac{1}{6}\bigg[4\int\limits_{\psi _{3}}w(x)\mathcal{F}\Big( \frac{w(x)+z(x)}{2w(x)}\Big)d\varrho (x)+\int\limits_{\psi _{3}}w(x)\mathcal{ F}\Big(\frac{z(x)}{w(x)}\Big)d\varrho (x)\bigg]-\int\limits_{\psi _{3}}w(x) \frac{\int\limits_{1}^{\frac{z(x)}{w(x)}}\mathcal{F}(t)dt}{\frac{z(x)}{w(x)} -1}d\varrho (x)\Big\vert \\ &\leq &\frac{(1+2^{\lambda +1})^{\frac{1}{\lambda }}}{4\big(3^{\lambda +1}(1+\lambda)\big)^{\frac{1}{\lambda }}}\bigg[\Big(\frac{3}{4}\Big)^{\frac{ 1}{\lambda }}+\Big(\frac{1}{4}\Big)^{\frac{1}{\lambda }}\bigg]\times \\ &&\Bigg[|\mathcal{F}^{\prime }(1)|\int\limits_{\psi _{3}}|z(x)-w(x)|d\varrho (x)+\int\limits_{\psi _{3}}|z(x)-w(x)|\big\vert\mathcal{F}^{\prime }\frac{ z(x)}{w(x)}\big\vert d\varrho (x)\Bigg]. \end{eqnarray}$

(3.5)

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:

$\begin{eqnarray} &&\Big\vert\frac{1}{6}\Big[\mathcal{D}_{\mathcal{F}}(w, z)+4\int\limits_{\psi }z(x)\mathcal{F}\Big(\frac{z(x)+w(x)}{2w(x)}d\varrho (x)\Big)\Big]-\mathcal{D }_{HH}^{\mathcal{F}}(w, z)\Big\vert \\ &\leq &\big(\frac{5}{18}\big)^{\frac{1}{\lambda }}\bigg[\Big(\frac{61}{324} \Big)^{\frac{1}{\lambda }}+\Big(\frac{29}{324}\Big)^{\frac{1}{\lambda }} \bigg]\times \\ &&\Bigg[|\mathcal{F}^{\prime }(1)|\int\limits_{\psi }\frac{|z(x)-w(x)|}{4} d\varrho (x)+\int\limits_{\psi }\frac{|z(x)-w(x)|}{4}\big\vert\mathcal{F} ^{\prime }\frac{z(x)}{w(x)}\big\vert d\varrho (x)\Bigg]. \end{eqnarray}$

(3.6)

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 $\mathfrak{G}$

$\begin{equation*} F(y) = P(Y\leq y) = \int\limits_{\varepsilon}^{x}\mathfrak{G}(t)dt. \end{equation*}$

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

$\begin{eqnarray*} &&\Big\vert \frac{1}{6}\Big[4P\bigg(Y\leq\frac{2\varepsilon+ \eta(\zeta, \varepsilon)}{2}\bigg)+1\Big]-\frac{1}{\eta(\zeta, \varepsilon)} \big(\varepsilon+\eta(\zeta, \varepsilon)-E(y)\big) \Big\vert \\ &&\leq\frac{\eta(\zeta, \varepsilon)}{4}\Big[\frac{1+2^{\mu+1}}{ 3^{\mu+1}(\mu+1)}\Big]^{\frac{1}{\mu}}\Big[\Big(\frac{3}{4}\Big)^{\frac{1}{ \lambda}}+ \Big(\frac{1}{4}\Big)^{\frac{1}{\lambda}}\Big]\Big[\vert \mathfrak{G}(\varepsilon)\vert+\vert \mathfrak{G}(\zeta)\vert\Big]. \end{eqnarray*}$

Proposition 3.4. By Corollary 2.3, we get the inequality

$\begin{eqnarray*} &&\Big\vert \frac{1}{6}\Big[4P\bigg(Y\leq\frac{2\varepsilon+ \eta(\zeta, \varepsilon)}{2}\bigg)+1\Big]-\frac{1}{\eta(\zeta, \varepsilon)} \big(\varepsilon+\eta(\zeta, \varepsilon)-E(y)\big) \Big\vert \\ &&\leq\frac{\eta(\zeta, \varepsilon)}{4}\big(\frac{5}{18}\big)^{1-\frac{1}{ \lambda}}\Big[\big(\frac{61}{324}\big)^{\frac{1}{\lambda}}+\big(\frac{29}{324 }\big)^{\frac{1}{\lambda}}\Big] \lbrack \vert \mathfrak{G} (\varepsilon)\vert+ \vert \mathfrak{G}(\zeta)\vert]. \end{eqnarray*}$

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.

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