Suggestion of a dynamic model of the development of neurodevelopmental disorders and the phenomenon of autism

Bodil Aggernæs; Bodil Aggernæs

doi:10.3934/molsci.2020008

AIMS Molecular Science

2020, Volume 7, Issue 2: 122-182. doi: 10.3934/molsci.2020008

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Research article Special Issues

Suggestion of a dynamic model of the development of neurodevelopmental disorders and the phenomenon of autism

Bodil Aggernæs ^{1,2
,
,}

1.
Department of Child and Adolescent Psychiatry, Psychiatry Region Zealand, 4000 Roskilde, Denmark; Department of Clinical Medicine, Faculty of Medical and Health Sciences, University of Copenhagen, 2200 Copenhagen N, Denmark
2.
Department of Adult Psychiatry, Mental Health Centre Northern Zealand, Mental Health Services of the Capital Region, 3400 Hillerød, Denmark

Received: 14 January 2020 Accepted: 27 May 2020 Published: 29 May 2020

The present article is a theoretical contribution suggesting a simple model of the dynamics involved in the development of neurodevelopmental disorders and the phenomenon of autism offering an explanation of why different individuals have different onset of manifest disease. The model relates to present genetic and epigenetic evidence and previous theoretical models. The dynamic model applies an individualized transdiagnostic and dimensional approach integrating several levels of influence involved in the dynamic interaction between an individual and their environment across time. The dynamic model illustrates the interaction between a basic neurobiological susceptibility, compensating mechanisms, and stress-related releasing mechanisms involved in the development of manifest clinical illness. The model has a particular focus on the dynamics of neurocognitive processes and their relationship to more basic information, psychological and social processes. A basic assumption guiding the model is that even quite normal events related to typical development may increase the risk of enduring stress in cognitively vulnerable individuals, further increasing their risk of developing manifest clinical illness. The model suggests that genetic variation, endogenous epigenetic processes of development, and epigenetic changes influenced by physical/chemical and social environmental risk and resilience factors all may contribute to phenotypical expression. A genetic susceptibility may translate into a biological susceptibility reflected in cognitive impairments. A cognitively vulnerable individual may be at increased risk of misinterpreting sensory inputs, potentially resulting in psychopathological expressions, e.g., autistic symptoms, observed across several neurodevelopmental disorders. The genetically influenced experience of an individual relates to cognitive phenomena occurring at the interfaces between the brain, mind, and society. The severity of clinical illness may differ from the severity of a disorder of reasoning. The dynamic model may help guide future development of personalized medicine in psychiatry and identify relevant points of intervention. A discussion of the implications of the model relating to epigenetic evidence and to previous theoretical models is included at the end of the paper.

Keywords:

Citation: Bodil Aggernæs. Suggestion of a dynamic model of the development of neurodevelopmental disorders and the phenomenon of autism[J]. AIMS Molecular Science, 2020, 7(2): 122-182. doi: 10.3934/molsci.2020008

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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.

Abbreviation ASD: Autism Spectrum Disorders; CNV: Copy Number Variants; DNA: Deoxyribonucleic Acid; ESSENCE: Early Symptomatic Syndromes Eliciting Neurodevelopmental Clinical Examinations; HPA axis: Hypothalamic-Pituitary-Adrenal Axis; IQ: Intelligence Quotient; LoF: Loss-of-Function; MAPP: Maltreatment-Associated Psychiatric Problems; OCD: Obsessive Compulsive Disorder; PTSD: Post Traumatic Stress Disorder; QoL: Quality of Life; RNA: Ribonucleic Acid (RNA); SNP: Single-Nucleotide Polymorphism; WGS: Whole-Genome Sequencing;
Acknowledgments

The author is very grateful to all the children, adolescents and their families who across the past years have shared their experiences as part of my clinical practice. Their experiences have inspired my theoretical research, thereby contributing to the development of the dynamic model, including the hypotheses guiding the model.

Conflict of interest

The author declares no conflicts of interest.

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