Parametric Inequalities for s-Convex Stochastic Processes via Caputo Fractional Derivatives

Abstract

This paper establishes a general parametric integral identity involving $(\mathfrak{n}+1)$-times differentiable stochastic processes, formulated entirely in terms of stochastic k-Caputo fractional derivatives. This identity serves as a unifying tool for deriving a broad class of parameter-dependent inequalities for differentiable s-convex stochastic processes. Remarkably, by assigning specific values to the underlying parameter, we have ensured our results specialize to well-known numerical integration inequalities, including those of midpoint, trapezium, Simpson, and Bullen types, in the stochastic fractional context. The findings not only enrich the theory of stochastic fractional calculus but also provide a flexible analytical apparatus for uncertainty quantification in fractional dynamical systems.

1. Introduction

Fractional calculus constitutes a canonical generalization of classical calculus, extending the operations of differentiation and integration to arbitrary real or complex orders. While derivatives of integer order encode only local information about a function, their fractional counterparts inherently incorporate global features, such as memory effects, hereditary mechanisms, and nonlocal dynamics, rendering them especially effective for describing complex systems that exhibit anomalous or history-dependent behavior [1,2,3]. Consequently, fractional-order models have been successfully employed across a broad spectrum of disciplines, including physics, engineering, economics, and biological sciences, often outperforming conventional integer-order approaches in capturing intricate real-world phenomena.

Among the several formulations proposed in the literature, the Caputo fractional derivative [4] has gained particular prominence, primarily because it allows for the natural incorporation of standard initial conditions in fractional differential equations [5,6].

Convexity is a cornerstone concept in mathematical analysis, especially in the study of approximation theory and error bounds for numerical integration rules. Its significance is vividly illustrated by the classical Hermite–Hadamard inequality, which establishes a precise relationship between the average value of a convex function over an interval and its values at the midpoint and endpoints, thereby serving as a fundamental tool in integral inequalities and convex analysis.

In [7], Farid et al. generalized the classical Hermite–Hadamard inequality to the setting of Caputo fractional derivatives for functions with convex n-th order derivatives Moreover, they derived a midpoint-type estimate for functions whose $(\mathfrak{n}+1)$th derivative has a convex absolute value.

To further enrich the fractional calculus framework, Farid et al. [8] introduced the notion of k-Caputo fractional derivatives, incorporating an auxiliary parameter $k>0$. This generalization offers enhanced adaptability in modeling phenomena where scaling effects or nonstandard normalization are relevant. Building on this extension, Farid et al. [9] established analogues of the Hermite–Hadamard and midpoint-type inequalities in the k-Caputo setting.

Stochastic processes, which constitute a fundamental pillar of probability theory, provide a rigorous framework for analyzing systems whose evolution is governed by randomness. Owing to these processes’ broad utility in physics, computer science, signal processing, cryptography, and control theory, this area has witnessed substantial theoretical and applied progress in recent decades. Indeed, stochastic processes have become essential for modeling dynamic phenomena characterized by uncertainty, thereby catalyzing innovations across both pure and applied disciplines [10,11].

The concept of convexity has naturally been extended to the stochastic setting. Building on this idea, Kotrys [12] established a stochastic analogue of the classical Hermite–Hadamard inequality. Recent research has further highlighted the pivotal role of generalized convexity notions in the analysis of stochastic processes through integral inequalities. For example, Materano et al. [13,14] derived Simpson-type inequalities for s-convex and quasi-convex stochastic processes, along with Ostrowski-type bounds for convex, s-convex, and quasi-convex cases. Agahi and Babakhani [15] extended classical Hermite–Hadamard and Jensen inequalities to the fractional setting for convex stochastic processes. Deng and Wang [16] contributed fractional Hermite–Hadamard inequalities tailored to $(\beta ,m)$-logarithmically convex functions, thereby expanding the reach of fractional calculus in stochastic contexts. Afzal et al. [17] investigated Jensen and Hermite–Hadamard inequalities for h-Godunova–Levin stochastic processes, underscoring the relevance of generalized convexity structures. Moreover, Meftah et al. [18] presented k-Riemann–Liouville fractional Maclaurin-type inequalities for s-convex stochastic processes, while Alzahrani et al. [19] developed Hermite–Hadamard-type estimates for harmonically s-convex stochastic processes. Other studies [20,21,22,23,24,25,26] have continued to deepen both the theoretical foundations and practical applications of convexity in stochastic analysis.

In [27], Rashid and coauthors introduced the notion of stochastic Caputo fractional derivatives, defined in the mean-square sense, and provided Hermite–Hadamard- and midpoint-type inequalities involving these stochastic fractional derivatives. More recently, Alruwaily et al. [28] introduced the notion of stochastic k-Caputo fractional derivatives and established Hermite–Hadamard, midpoint, and trapezium-type inequalities for stochastic convex processes.

Motivated by the research presented in [28] and the parametric studies conducted in [29,30,31], in this work, we generalize the existing framework of stochastic k-Caputo fractional derivatives for stochastic processes. Building upon these operators, we establish a unified parametric identity, which acts as the cornerstone for deriving a family of parameter-dependent inequalities tailored to $(\mathfrak{n}+1)$-times differentiable s-convex stochastic processes, expressed entirely through the newly defined stochastic k-Caputo fractional derivatives. Notably, by selecting particular values of the parameter, this general scheme recovers, as special cases, classical numerical integration inequalities of midpoint, trapezium, Simpson, and Bullen types.

The remainder of the paper is structured as follows. Section 2 recalls essential notions from stochastic calculus and fractional analysis required for our developments. In Section 3, we introduce a new parametrized identity, which forms the basis for proving a set of parametrized inequalities for mean-square differentiable s-convex stochastic processes in Section 4. Section 5 addresses specific instances obtained by choosing specific values of the parameter. A numerical illustration with graphical validation is provided in Section 6 to support the theoretical findings. Finally, Section 7 summarizes the main contributions and outlines potential directions for future research.

2. Preliminaries

This section outlines key notions that underpin the subsequent developments of our work.

Definition 1

([32]). Let $(\mathcal{W},\mathit{D},\mathcal{P})$ be a probability space. A random variable is a $\mathit{D}$-measurable mapping $\mathcal{P}:\mathcal{W}\to \mathbb{R}$. A function $\mathcal{P}:\mathcal{I}\times \mathcal{W}\to \mathbb{R}$, with $\mathcal{I}\subseteq \mathbb{R}$, is called a stochastic process if, for each fixed $\varkappa \in \mathcal{I}$, the mapping $\mathcal{P}(\varkappa ,·)$ is a random variable.

Definition 2

([32]). A stochastic process $\mathcal{P}:\mathcal{I}\times \mathcal{W}\to \mathbb{R}$ is said to be continuous in probability on $\mathcal{I}$ if, for every ${\varkappa }_0\in \mathcal{I}$,

$$\mathcal{P}(\varkappa ,·)\underset{\varkappa \to {\varkappa }_0}{\overset{\mathcal{P}}{\to }}\mathcal{P}({\varkappa }_0,·),$$

where $\stackrel{\mathcal{P}}{\to }$ denotes convergence in probability.

Definition 3

([32]). The process $\mathcal{P}$ is termed mean-square continuous on $\mathcal{I}$ if, for all ${\varkappa }_0\in \mathcal{I}$,

$$\underset{\varkappa \to {\varkappa }_0}{lim}\mathbb{E}\left[{\left(\mathcal{P}(\varkappa ,·)-\mathcal{P}({\varkappa }_0,·)\right)}^2\right]=0,$$

where $\mathbb{E}[·]$ denotes the expectation operator.

Definition 4

([32]). A stochastic process $\mathcal{P}$ is differentiable in probability at ${\varkappa }_0\in \mathcal{I}$ if there exists a random variable ${\mathcal{P}}^{\prime }({\varkappa }_0,·)$ such that

$${\displaystyle \frac{\mathcal{P}(\varkappa ,·)-\mathcal{P}({\varkappa }_0,·)}{\varkappa -{\varkappa }_0}}\underset{\varkappa \to {\varkappa }_0}{\overset{\mathcal{P}}{\to }}{\mathcal{P}}^{\prime }({\varkappa }_0,·).$$

Definition 5

([32]). The process $\mathcal{P}$ is mean-square-differentiable at ${\varkappa }_0\in \mathcal{I}$ if there exists a random variable ${\mathcal{P}}^{\prime }({\varkappa }_0,·)$ satisfying

$$\underset{\varkappa \to {\varkappa }_0}{lim}\mathbb{E}\left[{\left({\displaystyle \frac{\mathcal{P}(\varkappa ,·)-\mathcal{P}({\varkappa }_0,·)}{\varkappa -{\varkappa }_0}}-{\mathcal{P}}^{\prime }({\varkappa }_0,·)\right)}^2\right]=0.$$

Definition 6

([32]). Let $\mathcal{P}:\mathcal{I}\times \mathcal{W}\to \mathbb{R}$ be a stochastic process such that $\mathbb{E}\left[\mathcal{P}{(\varkappa ,·)}^2\right]<\infty$ for all $\varkappa \in \mathcal{I}$. The mean-square integral of $\mathcal{P}$ over $[{\mathfrak{e}}_1,{\mathfrak{e}}_2]\subseteq \mathcal{I}$ is the random variable $\mathcal{Q}:\mathcal{W}\to \mathbb{R}$ defined by the property that, for any sequence of partitions ${\mathfrak{e}}_1={\varkappa }_0<{\varkappa }_1<\dots <{\varkappa }_{\mathfrak{n}}={\mathfrak{e}}_2$ and any choice of sample points ${\wp }_k\in [{\varkappa }_{k-1},{\varkappa }_k]$, the Riemann sums converge in mean square to $\mathcal{Q}$, i.e.,

$$\underset{\parallel \mathsf{\Delta }\parallel \to 0}{lim}\mathbb{E}\left[{\left(\sum _{k=1}^{\mathfrak{n}}\mathcal{P}({\wp }_k,·)({\varkappa }_k-{\varkappa }_{k-1})-\mathcal{Q}(·)\right)}^2\right]=0,$$

where $\parallel \mathsf{\Delta }\parallel ={max}_k({\varkappa }_k-{\varkappa }_{k-1})$. In this case, we write

$$\mathcal{Q}(·)={\int }_{{\mathfrak{e}}_1}^{{\mathfrak{e}}_2}\mathcal{P}(\mathfrak{u},·)\mathrm{d}\mathfrak{u}(a.e.).$$

Definition 7

([33]). A stochastic process $\mathcal{P}:\mathcal{I}\times \mathcal{W}\to \mathbb{R}$ is called s-convex in the second sense if there exists $s\in (0,1]$ such that, for all $\mathfrak{u},\mathfrak{v}\in \mathcal{I}$ with $\mathfrak{u},\mathfrak{u}>0$ and every $\ell \in (0,1)$,

$$\mathcal{P}\left(\ell \mathfrak{u}+(1-\ell )\mathfrak{v},·\right)\le {\ell }^s\mathcal{P}(\mathfrak{u},·)+{(1-\ell )}^s\mathcal{P}(\mathfrak{v},·)(a.e.)$$

Definition 8

([28]). Let $\beta >0$, $k\ge 1$ and $\beta \notin \{1,2,3,·..\},\mathfrak{n}=\left[\beta \right]+1$. For the stochastic process $\mathcal{P}:\mathcal{I}\times \Omega \to \mathbb{R}$, the concept of stochastic mean-square Caputo k-fractional derivatives of the order β is defined as follows:

$$\left[{}_{ }{}^c\mathcal{D}_{{\mathfrak{e}}_1^{+}}^{\beta ,k}\mathcal{P}\right]\left(\varkappa \right)=\frac{1}{k{\mathsf{\Gamma }}_k\left(\mathfrak{n}-\frac{\beta }{k}\right)}\stackrel{\varkappa }{\underset{{\mathfrak{e}}_1}{\int }}{\left(\varkappa -\mathfrak{u}\right)}^{\mathfrak{n}-{\displaystyle \frac{\beta }{k}}-1}{\mathcal{P}}^{\left(\mathfrak{n}\right)}(\mathfrak{u},·)\mathrm{d}\mathfrak{u}(a.e.),\varkappa >{\mathfrak{e}}_1,$$

(1)

$$\left[{}_{ }{}^c\mathcal{D}_{{\mathfrak{e}}_2^{-}}^{\beta ,k}\mathcal{P}\right]\left(\varkappa \right)=\frac{{\left(-1\right)}^{\mathfrak{n}}}{k{\mathsf{\Gamma }}_k\left(\mathfrak{n}-\frac{\beta }{k}\right)}\stackrel{{\mathfrak{e}}_2}{\underset{\varkappa }{\int }}{\left(\mathfrak{u}-\varkappa \right)}^{\mathfrak{n}-{\displaystyle \frac{\beta }{k}}-1}{\mathcal{P}}^{\left(\mathfrak{n}\right)}(\mathfrak{u},·)\mathrm{d}\mathfrak{u}(a.e.),{\mathfrak{e}}_2>\varkappa ,$$

(2)

where ${\mathsf{\Gamma }}_k(·)$ denotes the k-gamma function defined in [34] as ${\mathsf{\Gamma }}_k\left(\delta \right)=\stackrel{\infty }{\underset{0}{\int }}{\mathfrak{u}}^{\delta -1}{e}^{-{\displaystyle \frac{{\mathfrak{u}}^k}{k}}}\mathrm{d}\mathfrak{u}$ satisfying ${\mathsf{\Gamma }}_k\left(\delta +k\right)=\delta {\mathsf{\Gamma }}_k\left(\delta \right)$.

3. Parametrized Identity for Stochastic k-Caputo Fractional Derivatives

This section introduces a new parametrized identity involving the introduced derivatives serving as a basis for the subsequent results.

Lemma 1.

Let $\mathcal{P}:\mathcal{I}\times \Omega \to \mathbb{R}$ be a mean-square-differentiable stochastic entity such that $\mathcal{P}\in A{C}^{\mathfrak{n}}\left[{\mathfrak{e}}_1,{\mathfrak{e}}_2\right]$, where ${\mathfrak{e}}_1,{\mathfrak{e}}_2\in {\mathcal{I}}^{\circ }$ with ${\mathfrak{e}}_1<{\mathfrak{e}}_2$. If ${\mathcal{P}}^{\left(\mathfrak{n}+1\right)}$ is mean-square-integrable on $\left[{\mathfrak{e}}_1,{\mathfrak{e}}_2\right]$, then the following equality holds almost everywhere:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\lambda }{2}}{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left({\mathfrak{e}}_1,·\right)+\left(1-\lambda \right){\mathcal{P}}^{\left(\mathfrak{n}\right)}\left(\frac{{\mathfrak{e}}_1+{\mathfrak{e}}_2}{2},·\right)+{\displaystyle \frac{\lambda }{2}}{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left({\mathfrak{e}}_2,·\right)-{\displaystyle \frac{2^{\mathfrak{n}-\frac{\beta }{k}-1}k{\mathsf{\Gamma }}_k\left(\mathfrak{n}-\frac{\beta }{k}+k\right)}{{\left({\mathfrak{e}}_2-{\mathfrak{e}}_1\right)}^{\mathfrak{n}-\frac{\beta }{k}}}}{\mathcal{C}}^{\beta ,k}\left[\mathcal{P}\right]\hfill \\ \hfill =& {\displaystyle \frac{{\mathfrak{e}}_2-{\mathfrak{e}}_1}{4}}\left(\underset{0}{\stackrel{1}{\int }}\left({\ell }^{\mathfrak{n}-{\displaystyle \frac{\beta }{k}}}-\lambda \right)\left({\mathcal{P}}^{\left(\mathfrak{n}+1\right)}\left(\left(1-\frac{\ell }{2}\right){\mathfrak{e}}_1+\frac{\ell }{2}{\mathfrak{e}}_2,·\right)-{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}\left(\frac{\ell }{2}{\mathfrak{e}}_1+\left(1-\frac{\ell }{2}\right){\mathfrak{e}}_2,·\right)\right)\mathrm{d}\ell \right)(a.e),\hfill \end{array}$$

where $\lambda \in [0,1]$, and

$${\mathcal{C}}^{\beta ,k}\left[\mathcal{P}\right]=\left({}_{ }{}^c\mathcal{D}_{\frac{{\mathfrak{e}}_1+{\mathfrak{e}}_2}{2}}^{\beta ,k}\left[\mathcal{P}\right]\left({\mathfrak{e}}_2\right)\right)+{\left(-1\right)}^{\mathfrak{n}}\left({}_{ }{}^c\mathcal{D}_{\frac{{\mathfrak{e}}_1+{\mathfrak{e}}_2}{2}}^{\beta ,k}\left[\mathcal{P}\right]\left({\mathfrak{e}}_1\right)\right).$$

(3)

Proof. 

Let

$${\mathcal{N}}_1=\underset{0}{\stackrel{{\displaystyle \frac{1}{2}}}{\int }}\left({\ell }^{\mathfrak{n}-{\displaystyle \frac{\beta }{k}}}-\lambda \right){\mathcal{P}}^{\left(\mathfrak{n}+1\right)}\left(\left(1-\frac{\ell }{2}\right){\mathfrak{e}}_1+\frac{\ell }{2}{\mathfrak{e}}_2,·\right)\mathrm{d}\ell$$

and

$${\mathcal{N}}_2=\underset{{\displaystyle \frac{1}{2}}}{\stackrel{1}{\int }}\left({\ell }^{\mathfrak{n}-{\displaystyle \frac{\beta }{k}}}-\lambda \right){\mathcal{P}}^{\left(\mathfrak{n}+1\right)}\left(\frac{\ell }{2}{\mathfrak{e}}_1+\left(1-\frac{\ell }{2}\right){\mathfrak{e}}_2,·\right)\mathrm{d}\ell .$$

By integrating, in part, ${\mathcal{N}}_1$ and then using the change in variable, we get

$$\begin{array}{cc}\hfill {\mathcal{N}}_1=& {\left.\frac{2}{{\mathfrak{e}}_2-{\mathfrak{e}}_1}\left({\ell }^{\mathfrak{n}-{\displaystyle \frac{\beta }{k}}}-\lambda \right){\mathcal{P}}^{\left(\mathfrak{n}\right)}\left(\left(1-\frac{\ell }{2}\right){\mathfrak{e}}_1+\frac{\ell }{2}{\mathfrak{e}}_2,·\right)\right|}_0^1-\frac{2}{{\mathfrak{e}}_2-{\mathfrak{e}}_1}\left(\mathfrak{n}-\frac{\beta }{k}\right)\underset{0}{\stackrel{1}{\int }}{\ell }^{\mathfrak{n}-{\displaystyle \frac{\beta }{k}}-1}{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left(\left(1-\frac{\ell }{2}\right){\mathfrak{e}}_1+\frac{\ell }{2}{\mathfrak{e}}_2,·\right)\mathrm{d}\ell \hfill \\ \hfill =& \frac{2\left(1-\lambda \right)}{{\mathfrak{e}}_2-{\mathfrak{e}}_1}{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left(\frac{{\mathfrak{e}}_1+{\mathfrak{e}}_2}{2},·\right)+\frac{2\lambda }{{\mathfrak{e}}_2-{\mathfrak{e}}_1}{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left({\mathfrak{e}}_1,·\right)-{\displaystyle \frac{2^{\mathfrak{n}-\frac{\beta }{k}+1}}{{\left({\mathfrak{e}}_2-{\mathfrak{e}}_1\right)}^{\mathfrak{n}-\frac{\beta }{k}-1}}}\left(\mathfrak{n}-\frac{\beta }{k}\right)\underset{{\mathfrak{e}}_1}{\stackrel{{\displaystyle \frac{{\mathfrak{e}}_1+{\mathfrak{e}}_2}{2}}}{\int }}{\left(\mathfrak{u}-{\mathfrak{e}}_1\right)}^{\mathfrak{n}-{\displaystyle \frac{\beta }{k}}-1}{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left(\mathfrak{u},·\right)\mathrm{d}\mathfrak{u}\hfill \\ \hfill =& \frac{2\left(1-\lambda \right)}{{\mathfrak{e}}_2-{\mathfrak{e}}_1}{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left(\frac{{\mathfrak{e}}_1+{\mathfrak{e}}_2}{2},·\right)+\frac{2\lambda }{{\mathfrak{e}}_2-{\mathfrak{e}}_1}{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left({\mathfrak{e}}_1,·\right)-{\left(-1\right)}^{\mathfrak{n}}{\displaystyle \frac{2^{\mathfrak{n}-\frac{\beta }{k}+1}k{\mathsf{\Gamma }}_k\left(\mathfrak{n}-\frac{\beta }{k}+k\right)}{{\left({\mathfrak{e}}_2-{\mathfrak{e}}_1\right)}^{\mathfrak{n}-\frac{\beta }{k}+1}}}\left({}_{ }{}^c\mathcal{D}_{\frac{{\mathfrak{e}}_1+{\mathfrak{e}}_2}{2}}^{\beta ,k}-\left[\mathcal{P}\right]\left({\mathfrak{e}}_1\right)\right),(\mathrm{a}.\mathrm{e}).\hfill \end{array}$$

(4)

Similarly, we obtain

$$\begin{array}{cc}\hfill {\mathcal{N}}_2=& {\left.-\frac{2}{{\mathfrak{e}}_2-{\mathfrak{e}}_1}\left({\ell }^{\mathfrak{n}-{\displaystyle \frac{\beta }{k}}}-\lambda \right){\mathcal{P}}^{\left(\mathfrak{n}\right)}\left(\frac{\ell }{2}{\mathfrak{e}}_1+\left(1-\frac{\ell }{2}\right){\mathfrak{e}}_2,·\right)\right|}_0^1+\frac{2\left(\mathfrak{n}-\frac{\beta }{k}\right)}{{\mathfrak{e}}_2-{\mathfrak{e}}_1}\underset{0}{\stackrel{1}{\int }}{\ell }^{\mathfrak{n}-{\displaystyle \frac{\beta }{k}}-1}{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left(\frac{\ell }{2}{\mathfrak{e}}_1+\left(1-\frac{\ell }{2}\right){\mathfrak{e}}_2,·\right)\mathrm{d}\ell \hfill \\ \hfill =& -\frac{2\left(1-\lambda \right)}{{\mathfrak{e}}_2-{\mathfrak{e}}_1}{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left(\frac{{\mathfrak{e}}_1+{\mathfrak{e}}_2}{2},·\right)-\frac{2\lambda }{{\mathfrak{e}}_2-{\mathfrak{e}}_1}\mathcal{P}\left({\mathfrak{e}}_2,·\right)+{\displaystyle \frac{2^{\mathfrak{n}-\frac{\beta }{k}+1}\left(\mathfrak{n}-\frac{\beta }{k}\right)}{{\left({\mathfrak{e}}_2-{\mathfrak{e}}_1\right)}^{\mathfrak{n}-\frac{\beta }{k}+1}}}\underset{{\displaystyle \frac{{\mathfrak{e}}_1+{\mathfrak{e}}_2}{2}}}{\stackrel{{\mathfrak{e}}_2}{\int }}{\left({\mathfrak{e}}_2-\mathfrak{u}\right)}^{\mathfrak{n}-{\displaystyle \frac{\beta }{k}}-1}{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left(\mathfrak{u},·\right)\mathrm{d}\mathfrak{u}\hfill \\ \hfill =& -\frac{2\left(1-\lambda \right)}{{\mathfrak{e}}_2-{\mathfrak{e}}_1}{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left(\frac{{\mathfrak{e}}_1+{\mathfrak{e}}_2}{2},·\right)-\frac{2\lambda }{{\mathfrak{e}}_2-{\mathfrak{e}}_1}\mathcal{P}\left({\mathfrak{e}}_2,·\right)+{\displaystyle \frac{2^{\mathfrak{n}-\beta +1}k\mathsf{\Gamma }\left(\mathfrak{n}-\frac{\beta }{k}+k\right)}{{\left({\mathfrak{e}}_2-{\mathfrak{e}}_1\right)}^{\mathfrak{n}-\frac{\beta }{k}+1}}}\left({}_{ }{}^c\mathcal{D}_{\frac{{\mathfrak{e}}_1+{\mathfrak{e}}_2}{2}}^{\beta ,k}+\left[\mathcal{P}\right]\left({\mathfrak{e}}_2\right)\right)(\mathrm{a}.\mathrm{e}).\hfill \end{array}$$

(5)

By subtracting (5) from (4) and then multiplying the resulting equality by ${\displaystyle \frac{{\mathfrak{e}}_2-{\mathfrak{e}}_1}{4}}$, we get the desired result. □

4. Parametrized Inequalities via Stochastic s-Convexity

Theorem 1.

Let $\mathcal{P}:\mathcal{I}\times \Omega \to \mathbb{R}$ be a mean-square-differentiable stochastic entity such that $\mathcal{P}\in A{C}^{\mathfrak{n}}\left[{\mathfrak{e}}_1,{\mathfrak{e}}_2\right]$, where ${\mathfrak{e}}_1,{\mathfrak{e}}_2\in {\mathcal{I}}^{\circ }$ with ${\mathfrak{e}}_1<{\mathfrak{e}}_2$ such that ${\mathcal{P}}^{\left(\mathfrak{n}+1\right)}$ is mean-square-integrable on $\left[{\mathfrak{e}}_1,{\mathfrak{e}}_2\right]$. If $\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}\right|$ is s-convex stochastic process, then the following inequality

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{\lambda }{2}}{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left({\mathfrak{e}}_1,·\right)+\left(1-\lambda \right){\mathcal{P}}^{\left(\mathfrak{n}\right)}\left(\frac{{\mathfrak{e}}_1+{\mathfrak{e}}_2}{2},·\right)+{\displaystyle \frac{\lambda }{2}}{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left({\mathfrak{e}}_2,·\right)-{\displaystyle \frac{2^{\mathfrak{n}-\frac{\beta }{k}-1}k{\mathsf{\Gamma }}_k\left(\mathfrak{n}-\frac{\beta }{k}+k\right)}{{\left({\mathfrak{e}}_2-{\mathfrak{e}}_1\right)}^{\mathfrak{n}-\frac{\beta }{k}}}}{\mathcal{C}}^{\beta ,k}\left[\mathcal{P}\right]\right|\hfill \\ \hfill \le & {\displaystyle \frac{{\mathfrak{e}}_2-{\mathfrak{e}}_1}{4}}\left({{\rm Y}}_1\left(s,\lambda \right)+{{\rm Y}}_2\left(s,\lambda \right)\right)\left(\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_1,·)\right|+\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_2,·)\right|\right)\hfill \end{array}$$

holds almost everywhere, where ${\mathcal{C}}^{\beta ,k}\left[\mathcal{P}\right]$ is defined as in (3), and

$$\begin{array}{cc}\hfill {{\rm Y}}_1\left(s,\lambda \right)=\underset{0}{\stackrel{1}{\int }}\left|{\ell }^{\mathfrak{n}-{\displaystyle \frac{\beta }{k}}}-\lambda \right|{\left(1-\frac{\ell }{2}\right)}^s\mathrm{d}\ell =& {\displaystyle \frac{\lambda }{s+1}}\left(2+{\left(\frac{1}{2}\right)}^s\left(1-2{\left(2-{\lambda }^{{\displaystyle \frac{k}{\mathfrak{n}k-\beta }}}\right)}^{s+1}\right)\right)\hfill \\ \hfill & +2^{\mathfrak{n}-{\displaystyle \frac{\beta }{k}}+1}{B}_{{\displaystyle \frac{1}{2}}}\left(\mathfrak{n}-\frac{\beta }{k}+1,s+1\right)\hfill \\ \hfill & -2^{\mathfrak{n}-{\displaystyle \frac{\beta }{k}}+2}{B}_{{\displaystyle \frac{{\lambda }^{\frac{k}{\mathfrak{n}k-\beta }}}{2}}}\left(\mathfrak{n}-\frac{\beta }{k}+1,s+1\right)\hfill \end{array}$$

(6)

and

$${{\rm Y}}_2\left(s,\lambda \right)=\underset{0}{\stackrel{1}{\int }}\left|{\ell }^{\mathfrak{n}-{\displaystyle \frac{\beta }{k}}}-\lambda \right|{\left(\frac{\ell }{2}\right)}^s\mathrm{d}\ell ={\displaystyle \frac{\left(k\mathfrak{n}-\beta \right){\lambda }^{\frac{\left(\mathfrak{n}+s+1\right)k-\beta }{\mathfrak{n}k-\beta }}}{2^{s-1}\left(s+1\right)\left(\left(\mathfrak{n}+s+1\right)k-\beta \right)}}+{\displaystyle \frac{k\left(s+1\right)-\lambda \left(\left(\mathfrak{n}+s+1\right)k-\beta \right)}{2^s\left(\left(\mathfrak{n}+s+1\right)k-\beta \right)\left(s+1\right)}},$$

(7)

where $B_{\sigma }$ denotes the incomplete Beta function given by

$$B_{\sigma }(\mathfrak{p},\mathfrak{q})=\underset{0}{\stackrel{\sigma }{\int }}{\ell }^{\mathfrak{p}-1}{(1-\ell )}^{\mathfrak{q}-1}\mathrm{d}\ell ,0\le \sigma \le 1.$$

Proof. 

Using Lemma 1, and the fact that $\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}\right|$ is an s-convex stochastic process, we deduce that

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{\lambda }{2}}{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left({\mathfrak{e}}_1,·\right)+\left(1-\lambda \right){\mathcal{P}}^{\left(\mathfrak{n}\right)}\left(\frac{{\mathfrak{e}}_1+{\mathfrak{e}}_2}{2},·\right)+{\displaystyle \frac{\lambda }{2}}{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left({\mathfrak{e}}_2,·\right)-{\displaystyle \frac{2^{\mathfrak{n}-\frac{\beta }{k}-1}k{\mathsf{\Gamma }}_k\left(\mathfrak{n}-\frac{\beta }{k}+k\right)}{{\left({\mathfrak{e}}_2-{\mathfrak{e}}_1\right)}^{\mathfrak{n}-\frac{\beta }{k}}}}{\mathcal{C}}^{\beta ,k}\left[\mathcal{P}\right]\right|\hfill \\ \hfill \le & {\displaystyle \frac{{\mathfrak{e}}_2-{\mathfrak{e}}_1}{4}}\left(\underset{0}{\stackrel{1}{\int }}\left|{\ell }^{\mathfrak{n}-{\displaystyle \frac{\beta }{k}}}-\lambda \right|\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}\left(\left(1-\frac{\ell }{2}\right){\mathfrak{e}}_1+\frac{\ell }{2}{\mathfrak{e}}_2,·\right)\right|\mathrm{d}\ell +\underset{0}{\stackrel{1}{\int }}\left|{\ell }^{\mathfrak{n}-{\displaystyle \frac{\beta }{k}}}-\lambda \right|\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}\left(\frac{\ell }{2}{\mathfrak{e}}_1+\left(1-\frac{\ell }{2}\right){\mathfrak{e}}_2,·\right)\right|\mathrm{d}\ell \right)\hfill \\ \hfill \le & {\displaystyle \frac{{\mathfrak{e}}_2-{\mathfrak{e}}_1}{4}}\left(\underset{0}{\stackrel{1}{\int }}\left|{\ell }^{\mathfrak{n}-{\displaystyle \frac{\beta }{k}}}-\lambda \right|\left({\left(1-\frac{\ell }{2}\right)}^s\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_1,·)\right|+{\left(\frac{\ell }{2}\right)}^s\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_2,·)\right|\right)\mathrm{d}\ell \right.\hfill \\ \hfill & +\left.\underset{0}{\stackrel{1}{\int }}\left|{\ell }^{\mathfrak{n}-{\displaystyle \frac{\beta }{k}}}-\lambda \right|\left({\left(\frac{\ell }{2}\right)}^s\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_1,·)\right|+{\left(1-\frac{\ell }{2}\right)}^s\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_2,·)\right|\right)\mathrm{d}\ell \right)\hfill \\ \hfill =& {\displaystyle \frac{{\mathfrak{e}}_2-{\mathfrak{e}}_1}{4}}\left(\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_1,·)\right|+\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_2,·)\right|\right)\underset{0}{\stackrel{1}{\int }}\left|{\ell }^{\mathfrak{n}-{\displaystyle \frac{\beta }{k}}}-\lambda \right|\left({\left(1-\frac{\ell }{2}\right)}^s+{\left(\frac{\ell }{2}\right)}^s\right)\mathrm{d}\ell \hfill \\ \hfill =& {\displaystyle \frac{{\mathfrak{e}}_2-{\mathfrak{e}}_1}{4}}\left({{\rm Y}}_1\left(s,\lambda \right)+{{\rm Y}}_2\left(s,\lambda \right)\right)\left(\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_1,·)\right|+\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_2,·)\right|\right),\hfill \end{array}$$

where we have used (6) and (7).

This completes the proof. □

Corollary 1.

If we attempt to make $s=1$, Theorem 1 yields the following parametrized inequality via stochastic convexity

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{\lambda }{2}}{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left({\mathfrak{e}}_1,·\right)+\left(1-\lambda \right){\mathcal{P}}^{\left(\mathfrak{n}\right)}\left(\frac{{\mathfrak{e}}_1+{\mathfrak{e}}_2}{2},·\right)+{\displaystyle \frac{\lambda }{2}}{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left({\mathfrak{e}}_2,·\right)-{\displaystyle \frac{2^{\mathfrak{n}-\frac{\beta }{k}-1}k{\mathsf{\Gamma }}_k\left(\mathfrak{n}-\frac{\beta }{k}+k\right)}{{\left({\mathfrak{e}}_2-{\mathfrak{e}}_1\right)}^{\mathfrak{n}-\frac{\beta }{k}}}}{\mathcal{C}}^{\beta ,k}\left[\mathcal{P}\right]\right|\hfill \\ \hfill \le & {\displaystyle \frac{{\mathfrak{e}}_2-{\mathfrak{e}}_1}{4}}\left(\frac{2\left(k\mathfrak{n}-\beta \right){\lambda }^{\frac{k\left(\mathfrak{n}+1\right)-\beta }{\mathfrak{n}k-\beta }}-\left(k\left(\mathfrak{n}+1\right)-\beta \right)\lambda +k}{\left(k\left(\mathfrak{n}+1\right)-\beta \right)}\right)\left(\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_1,·)\right|+\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_2,·)\right|\right)(a.e.),\hfill \end{array}$$

where ${\mathcal{C}}^{\beta ,k}\left[\mathcal{P}\right]$ is defined as in (3), and we have used the facts that

$${{\rm Y}}_1\left(1,\lambda \right)={\displaystyle \frac{2\left(k\mathfrak{n}-\beta \right)}{k\left(\mathfrak{n}+1\right)-\beta }}{\lambda }^{\frac{k\left(\mathfrak{n}+1\right)-\beta }{\mathfrak{n}k-\beta }}-{\displaystyle \frac{k\mathfrak{n}-\beta }{2\left(k\left(\mathfrak{n}+2\right)-\beta \right)}}{\lambda }^{\frac{k\left(\mathfrak{n}+2\right)-\beta }{\mathfrak{n}k-\beta }}-{\displaystyle \frac{2k+3\lambda \left(k\left(\mathfrak{n}+2\right)-\beta \right)}{4\left(k\left(\mathfrak{n}+2\right)-\beta \right)}}+{\displaystyle \frac{k}{k\left(\mathfrak{n}+1\right)-\beta }}$$

(8)

and

$${{\rm Y}}_2\left(1,\lambda \right)={\displaystyle \frac{k\mathfrak{n}-\beta }{2\left(k\left(\mathfrak{n}+2\right)-\beta \right)}}{\lambda }^{\frac{\left(\mathfrak{n}+2\right)k-\beta }{\mathfrak{n}k-\beta }}+{\displaystyle \frac{2k-\lambda \left(\left(\mathfrak{n}+2\right)k-\beta \right)}{4\left(k\left(\mathfrak{n}+2\right)-\beta \right)}}.$$

(9)

Theorem 2.

Let $\mathcal{P}:\mathcal{I}\times \Omega \to \mathbb{R}$ be a mean-square-differentiable stochastic such that $\mathcal{P}\in A{C}^{\mathfrak{n}}\left[{\mathfrak{e}}_1,{\mathfrak{e}}_2\right]$, where ${\mathfrak{e}}_1,{\mathfrak{e}}_2\in {\mathcal{I}}^{\circ }$ with ${\mathfrak{e}}_1<{\mathfrak{e}}_2$ such that ${\mathcal{P}}^{\left(\mathfrak{n}+1\right)}$ is mean-square-integrable on $\left[{\mathfrak{e}}_1,{\mathfrak{e}}_2\right]$. If ${\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}\right|}^q$ is an s-convex stochastic process for $q>1$ with $pq=p+q$, then the following inequality

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{\lambda }{2}}{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left({\mathfrak{e}}_1,·\right)+\left(1-\lambda \right){\mathcal{P}}^{\left(\mathfrak{n}\right)}\left(\frac{{\mathfrak{e}}_1+{\mathfrak{e}}_2}{2},·\right)+{\displaystyle \frac{\lambda }{2}}{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left({\mathfrak{e}}_2,·\right)-{\displaystyle \frac{2^{\mathfrak{n}-\frac{\beta }{k}-1}k{\mathsf{\Gamma }}_k\left(\mathfrak{n}-\frac{\beta }{k}+k\right)}{{\left({\mathfrak{e}}_2-{\mathfrak{e}}_1\right)}^{\mathfrak{n}-\frac{\beta }{k}}}}{\mathcal{C}}^{\beta ,k}\left[\mathcal{P}\right]\right|\hfill \\ \hfill \le & {\displaystyle \frac{{\mathfrak{e}}_2-{\mathfrak{e}}_1}{4}}{\left({\displaystyle \frac{k{\lambda }^{p+\frac{k}{k\mathfrak{n}-\beta }}}{k\mathfrak{n}-\beta }}B\left({\displaystyle \frac{k}{k\mathfrak{n}-\beta }},p+1\right)+{\displaystyle \frac{k{\left(1-\lambda \right)}^{p+1}}{(k\mathfrak{n}-\beta )(p+1)}}{}_2{}^{ }F_1^{ }\left({\displaystyle \frac{k\left(\mathfrak{n}-1\right)-\beta }{k\mathfrak{n}-\beta }},1,p+2;1-\lambda \right)\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & \times \left({\left({\displaystyle \frac{\left(2^{s+1}-1\right){\left|{\mathcal{P}}^{\prime }({\mathfrak{e}}_1,·)\right|}^q+{\left|{\mathcal{P}}^{\prime }({\mathfrak{e}}_2,·)\right|}^q}{2^s\left(s+1\right)}}\right)}^{{\displaystyle \frac{1}{q}}}+{\left({\displaystyle \frac{{\left|{\mathcal{P}}^{\prime }({\mathfrak{e}}_1,·)\right|}^q+\left(2^{s+1}-1\right){\left|{\mathcal{P}}^{\prime }({\mathfrak{e}}_2,·)\right|}^q}{2^s\left(s+1\right)}}\right)}^{{\displaystyle \frac{1}{q}}}\right)\hfill \end{array}$$

holds almost everywhere, where ${\mathcal{C}}^{\beta ,k}\left[\mathcal{P}\right]$ is defined as in (3), $B\left(·,·\right)$ and ${}_2{}^{ }F_1^{ }\left(·,·,·;·\right)$ are beta and hypergeometric functions, respectively.

Proof. 

Using Lemma 1, Hölder’s inequality, and the fact that ${\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}\right|}^q$ is an s-convex stochastic process, we deduce that

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{\lambda }{2}}{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left({\mathfrak{e}}_1,·\right)+\left(1-\lambda \right){\mathcal{P}}^{\left(\mathfrak{n}\right)}\left(\frac{{\mathfrak{e}}_1+{\mathfrak{e}}_2}{2},·\right)+{\displaystyle \frac{\lambda }{2}}{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left({\mathfrak{e}}_2,·\right)-{\displaystyle \frac{2^{\mathfrak{n}-\frac{\beta }{k}-1}k{\mathsf{\Gamma }}_k\left(\mathfrak{n}-\frac{\beta }{k}+k\right)}{{\left({\mathfrak{e}}_2-{\mathfrak{e}}_1\right)}^{\mathfrak{n}-\frac{\beta }{k}}}}{\mathcal{C}}^{\beta ,k}\left[\mathcal{P}\right]\right|\hfill \\ \hfill \le & {\displaystyle \frac{{\mathfrak{e}}_2-{\mathfrak{e}}_1}{4}}\left({\left(\underset{0}{\stackrel{1}{\int }}{\left|{\ell }^{\mathfrak{n}-{\displaystyle \frac{\beta }{k}}}-\lambda \right|}^p\mathrm{d}\ell \right)}^{{\displaystyle \frac{1}{p}}}{\left(\underset{0}{\stackrel{1}{\int }}{\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}\left(\left(1-\frac{\ell }{2}\right){\mathfrak{e}}_1+\frac{\ell }{2}{\mathfrak{e}}_2,·\right)\right|}^q\mathrm{d}\ell \right)}^{{\displaystyle \frac{1}{q}}}\right.\hfill \\ \hfill & +\left.{\left(\underset{0}{\stackrel{1}{\int }}{\left|{\ell }^{\mathfrak{n}-{\displaystyle \frac{\beta }{k}}}-\lambda \right|}^p\mathrm{d}\ell \right)}^{{\displaystyle \frac{1}{p}}}{\left(\underset{0}{\stackrel{1}{\int }}{\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}\left(\frac{\ell }{2}{\mathfrak{e}}_1+\left(1-\frac{\ell }{2}\right){\mathfrak{e}}_2,·\right)\right|}^q\mathrm{d}\ell \right)}^{{\displaystyle \frac{1}{q}}}\right)\hfill \\ \hfill \le & {\displaystyle \frac{{\mathfrak{e}}_2-{\mathfrak{e}}_1}{4}}{\left(\underset{0}{\stackrel{1}{\int }}{\left|{\ell }^{\mathfrak{n}-{\displaystyle \frac{\beta }{k}}}-\lambda \right|}^p\mathrm{d}\ell \right)}^{{\displaystyle \frac{1}{p}}}\left({\left(\underset{0}{\stackrel{1}{\int }}\left({\left(1-\frac{\ell }{2}\right)}^s{\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_1,·)\right|}^q+{\left(\frac{\ell }{2}\right)}^s{\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_2,·)\right|}^q\right)\mathrm{d}\ell \right)}^{{\displaystyle \frac{1}{q}}}\right.\hfill \\ \hfill & +\left.{\left(\underset{0}{\stackrel{1}{\int }}\left({\left(\frac{\ell }{2}\right)}^s{\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_1,·)\right|}^q+{\left(1-\frac{\ell }{2}\right)}^s{\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_2,·)\right|}^q\right)\mathrm{d}\ell \right)}^{{\displaystyle \frac{1}{q}}}\right)\hfill \\ \hfill =& {\displaystyle \frac{{\mathfrak{e}}_2-{\mathfrak{e}}_1}{4}}{\left({\displaystyle \frac{k{\lambda }^{p+\frac{k}{k\mathfrak{n}-\beta }}}{k\mathfrak{n}-\beta }}B\left({\displaystyle \frac{k}{k\mathfrak{n}-\beta }},p+1\right)+{\displaystyle \frac{k{\left(1-\lambda \right)}^{p+1}}{(k\mathfrak{n}-\beta )(p+1)}}{}_2{}^{ }F_1^{ }\left({\displaystyle \frac{k\left(\mathfrak{n}-1\right)-\beta }{k\mathfrak{n}-\beta }},1,p+2;1-\lambda \right)\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & \times \left({\left({\displaystyle \frac{\left(2^{s+1}-1\right){\left|{\mathcal{P}}^{\prime }({\mathfrak{e}}_1,·)\right|}^q+{\left|{\mathcal{P}}^{\prime }({\mathfrak{e}}_2,·)\right|}^q}{2^s\left(s+1\right)}}\right)}^{{\displaystyle \frac{1}{q}}}+{\left({\displaystyle \frac{{\left|{\mathcal{P}}^{\prime }({\mathfrak{e}}_1,·)\right|}^q+\left(2^{s+1}-1\right){\left|{\mathcal{P}}^{\prime }({\mathfrak{e}}_2,·)\right|}^q}{2^s\left(s+1\right)}}\right)}^{{\displaystyle \frac{1}{q}}}\right).\hfill \end{array}$$

Thus, the proof is completed. □

Corollary 2.

In Theorem 2, if we assume $s=1$, then we get

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{\lambda }{2}}{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left({\mathfrak{e}}_1,·\right)+\left(1-\lambda \right){\mathcal{P}}^{\left(\mathfrak{n}\right)}\left(\frac{{\mathfrak{e}}_1+{\mathfrak{e}}_2}{2},·\right)+{\displaystyle \frac{\lambda }{2}}{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left({\mathfrak{e}}_2,·\right)-{\displaystyle \frac{2^{\mathfrak{n}-\frac{\beta }{k}-1}k{\mathsf{\Gamma }}_k\left(\mathfrak{n}-\frac{\beta }{k}+k\right)}{{\left({\mathfrak{e}}_2-{\mathfrak{e}}_1\right)}^{\mathfrak{n}-\frac{\beta }{k}}}}{\mathcal{C}}^{\beta ,k}\left[\mathcal{P}\right]\right|\hfill \\ \hfill \le & {\displaystyle \frac{{\mathfrak{e}}_2-{\mathfrak{e}}_1}{4}}{\left({\displaystyle \frac{k{\lambda }^{p+\frac{k}{k\mathfrak{n}-\beta }}}{k\mathfrak{n}-\beta }}B\left(\frac{k}{k\mathfrak{n}-\beta },p+1\right)+{\displaystyle \frac{k{\left(1-\lambda \right)}^{p+1}}{(k\mathfrak{n}-\beta )(p+1)}}{}_2{}^{ }F_1^{ }\left({\displaystyle \frac{k\left(\mathfrak{n}-1\right)-\beta }{k\mathfrak{n}-\beta }},1,p+2;1-\lambda \right)\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & \times \left({\left({\displaystyle \frac{3{\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_1,·)\right|}^q+{\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_2,·)\right|}^q}{4}}\right)}^{{\displaystyle \frac{1}{q}}}+{\left({\displaystyle \frac{{\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_1,·)\right|}^q+3{\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_2,·)\right|}^q}{4}}\right)}^{{\displaystyle \frac{1}{q}}}\right)(a.e.).\hfill \end{array}$$

Theorem 3.

Let $\mathcal{P}:\mathcal{I}\times \Omega \to \mathbb{R}$ be a mean-square-differentiable stochastic process such that $\mathcal{P}\in A{C}^{\mathfrak{n}}\left[{\mathfrak{e}}_1,{\mathfrak{e}}_2\right]$ where ${\mathfrak{e}}_1,{\mathfrak{e}}_2\in {\mathcal{I}}^{\circ }$ with ${\mathfrak{e}}_1<{\mathfrak{e}}_2$ such that ${\mathcal{P}}^{\left(\mathfrak{n}+1\right)}$ is mean-square-integrable on $\left[{\mathfrak{e}}_1,{\mathfrak{e}}_2\right]$. If ${\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}\right|}^q$ is an s-convex stochastic process for $q>1$, then the following inequality

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{\lambda }{2}}{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left({\mathfrak{e}}_1,·\right)+\left(1-\lambda \right){\mathcal{P}}^{\left(\mathfrak{n}\right)}\left(\frac{{\mathfrak{e}}_1+{\mathfrak{e}}_2}{2},·\right)+{\displaystyle \frac{\lambda }{2}}{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left({\mathfrak{e}}_2,·\right)-{\displaystyle \frac{2^{\mathfrak{n}-\frac{\beta }{k}-1}k{\mathsf{\Gamma }}_k\left(\mathfrak{n}-\frac{\beta }{k}+k\right)}{{\left({\mathfrak{e}}_2-{\mathfrak{e}}_1\right)}^{\mathfrak{n}-\frac{\beta }{k}}}}{\mathcal{C}}^{\beta ,k}\left[\mathcal{P}\right]\right|\hfill \\ \hfill \le & {\displaystyle \frac{{\mathfrak{e}}_2-{\mathfrak{e}}_1}{4}}{\left({\displaystyle \frac{k}{k\left(\mathfrak{n}+1\right)-\beta }}-\lambda +{\displaystyle \frac{2\left(k\mathfrak{n}-\beta \right)}{k\left(\mathfrak{n}+1\right)-\beta }}{\lambda }^{{\displaystyle \frac{k\left(\mathfrak{n}+1\right)-\beta }{k\mathfrak{n}-\beta }}}\right)}^{1-{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & \times \left({\left({{\rm Y}}_1\left(s,\lambda \right){\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_1,·)\right|}^q+{{\rm Y}}_2\left(s,\lambda \right){\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_2,·)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\right.\hfill \\ \hfill & +\left.{\left({{\rm Y}}_2\left(s,\lambda \right){\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_1,·)\right|}^q+{{\rm Y}}_1\left(s,\lambda \right){\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_2,·)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\right)\hfill \end{array}$$

holds almost everywhere, where ${\mathcal{C}}^{\beta ,k}\left[\mathcal{P}\right]$ is defined as in (3) and ${{\rm Y}}_1$ and ${{\rm Y}}_2$ are defined as in (6) and (7), respectively.

Proof. 

Using Lemma 1, power mean inequality, and the fact that ${\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}\right|}^q$ is an s-convex stochastic process, we deduce that

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{\lambda }{2}}{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left({\mathfrak{e}}_1,·\right)+\left(1-\lambda \right){\mathcal{P}}^{\left(\mathfrak{n}\right)}\left(\frac{{\mathfrak{e}}_1+{\mathfrak{e}}_2}{2},·\right)+{\displaystyle \frac{\lambda }{2}}{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left({\mathfrak{e}}_2,·\right)-{\displaystyle \frac{2^{\mathfrak{n}-\frac{\beta }{k}-1}k{\mathsf{\Gamma }}_k\left(\mathfrak{n}-\frac{\beta }{k}+k\right)}{{\left({\mathfrak{e}}_2-{\mathfrak{e}}_1\right)}^{\mathfrak{n}-\frac{\beta }{k}}}}{\mathcal{C}}^{\beta ,k}\left[\mathcal{P}\right]\right|\hfill \\ \hfill \le & {\displaystyle \frac{{\mathfrak{e}}_2-{\mathfrak{e}}_1}{4}}\left({\left(\underset{0}{\stackrel{1}{\int }}\left|{\ell }^{\mathfrak{n}-{\displaystyle \frac{\beta }{k}}}-\lambda \right|\mathrm{d}\ell \right)}^{1-{\displaystyle \frac{1}{q}}}{\left(\underset{0}{\stackrel{1}{\int }}\left|{\ell }^{\mathfrak{n}-{\displaystyle \frac{\beta }{k}}}-\lambda \right|{\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}\left(\left(1-\frac{\ell }{2}\right){\mathfrak{e}}_1+\frac{\ell }{2}{\mathfrak{e}}_2,·\right)\right|}^q\mathrm{d}\ell \right)}^{{\displaystyle \frac{1}{q}}}\right.\hfill \\ \hfill & +\left.{\left(\underset{0}{\stackrel{1}{\int }}\left|{\ell }^{\mathfrak{n}-{\displaystyle \frac{\beta }{k}}}-\lambda \right|\mathrm{d}\ell \right)}^{1-{\displaystyle \frac{1}{q}}}{\left(\underset{0}{\stackrel{1}{\int }}\left|{\ell }^{\mathfrak{n}-{\displaystyle \frac{\beta }{k}}}-\lambda \right|{\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}\left(\frac{\ell }{2}{\mathfrak{e}}_1+\left(1-\frac{\ell }{2}\right){\mathfrak{e}}_2,·\right)\right|}^q\mathrm{d}\ell \right)}^{{\displaystyle \frac{1}{q}}}\right)\hfill \\ \hfill \le & {\displaystyle \frac{{\mathfrak{e}}_2-{\mathfrak{e}}_1}{4}}{\left(\underset{0}{\stackrel{1}{\int }}\left|{\ell }^{\mathfrak{n}-{\displaystyle \frac{\beta }{k}}}-\lambda \right|\mathrm{d}\ell \right)}^{1-{\displaystyle \frac{1}{q}}}\left({\left(\underset{0}{\stackrel{1}{\int }}\left|{\ell }^{\mathfrak{n}-{\displaystyle \frac{\beta }{k}}}-\lambda \right|\left({\left(1-\frac{\ell }{2}\right)}^s{\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_1,·)\right|}^q+{\left(\frac{\ell }{2}\right)}^s{\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_2,·)\right|}^q\right)\mathrm{d}\ell \right)}^{{\displaystyle \frac{1}{q}}}\right.\hfill \\ \hfill & +\left.{\left(\underset{0}{\stackrel{1}{\int }}\left|{\ell }^{\mathfrak{n}-{\displaystyle \frac{\beta }{k}}}-\lambda \right|\left({\left(\frac{\ell }{2}\right)}^s{\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_1,·)\right|}^q+{\left(1-\frac{\ell }{2}\right)}^s{\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_2,·)\right|}^q\right)\mathrm{d}\ell \right)}^{{\displaystyle \frac{1}{q}}}\right)\hfill \\ \hfill =& {\displaystyle \frac{{\mathfrak{e}}_2-{\mathfrak{e}}_1}{4}}{\left({\displaystyle \frac{k}{k\left(\mathfrak{n}+1\right)-\beta }}-\lambda +{\displaystyle \frac{2\left(k\mathfrak{n}-\beta \right)}{k\left(\mathfrak{n}+1\right)-\beta }}{\lambda }^{{\displaystyle \frac{k\left(\mathfrak{n}+1\right)-\beta }{k\mathfrak{n}-\beta }}}\right)}^{1-{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & \times \left({\left({{\rm Y}}_1\left(s,\lambda \right){\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_1,·)\right|}^q+{{\rm Y}}_2\left(s,\lambda \right){\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_2,·)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\right.\hfill \\ \hfill & +\left.{\left({{\rm Y}}_2\left(s,\lambda \right){\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_1,·)\right|}^q+{{\rm Y}}_1\left(s,\lambda \right){\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_2,·)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\right),\hfill \end{array}$$

where ${{\rm Y}}_1$ and ${{\rm Y}}_2$ are defined as in (6) and (7), respectively.

This completes the proof. □

Corollary 3.

If we attempt to make $s=1$, Theorem 3 yields the following parametrized inequality via stochastic convexity:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{\lambda }{2}}{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left({\mathfrak{e}}_1,·\right)+\left(1-\lambda \right){\mathcal{P}}^{\left(\mathfrak{n}\right)}\left(\frac{{\mathfrak{e}}_1+{\mathfrak{e}}_2}{2},·\right)+{\displaystyle \frac{\lambda }{2}}{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left({\mathfrak{e}}_2,·\right)-{\displaystyle \frac{2^{\mathfrak{n}-\frac{\beta }{k}-1}k{\mathsf{\Gamma }}_k\left(\mathfrak{n}-\frac{\beta }{k}+k\right)}{{\left({\mathfrak{e}}_2-{\mathfrak{e}}_1\right)}^{\mathfrak{n}-\frac{\beta }{k}}}}{\mathcal{C}}^{\beta ,k}\left[\mathcal{P}\right]\right|\hfill \\ \hfill \le & {\displaystyle \frac{{\mathfrak{e}}_2-{\mathfrak{e}}_1}{4}}{\left({\displaystyle \frac{k}{k\left(\mathfrak{n}+1\right)-\beta }}-\lambda +{\displaystyle \frac{2\left(k\mathfrak{n}-\beta \right)}{k\left(\mathfrak{n}+1\right)-\beta }}{\lambda }^{{\displaystyle \frac{k\left(\mathfrak{n}+1\right)-\beta }{k\mathfrak{n}-\beta }}}\right)}^{1-{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & \times \left({\left({{\rm Y}}_1\left(1,\lambda \right){\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_1,·)\right|}^q+{{\rm Y}}_2\left(1,\lambda \right){\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_2,·)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\right.\hfill \\ \hfill & +\left.{\left({{\rm Y}}_2\left(1,\lambda \right){\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_1,·)\right|}^q+{{\rm Y}}_1\left(1,\lambda \right){\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_2,·)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\right)(a.e.),\hfill \end{array}$$

where ${{\rm Y}}_1\left(1,\lambda \right)$ and ${{\rm Y}}_2\left(1,\lambda \right)$ are defined as in (8) and (9), respectively.

5. Special Instances

5.1. Midpoint-Type Inequalities

Corollary 4.

If we set $\lambda =0$, Theorem 1 yields the following midpoint inequality via stochastic s-convexity:

$$\begin{array}{cc}\hfill & \left|{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left(\frac{{\mathfrak{e}}_1+{\mathfrak{e}}_2}{2},·\right)-{\displaystyle \frac{2^{\mathfrak{n}-\frac{\beta }{k}-1}k{\mathsf{\Gamma }}_k\left(\mathfrak{n}-\frac{\beta }{k}+k\right)}{{\left({\mathfrak{e}}_2-{\mathfrak{e}}_1\right)}^{\mathfrak{n}-\frac{\beta }{k}}}}{\mathcal{C}}^{\beta ,k}\left[\mathcal{P}\right]\right|\hfill \\ \hfill \le & {\displaystyle \frac{{\mathfrak{e}}_2-{\mathfrak{e}}_1}{4}}\left(2^{\mathfrak{n}-{\displaystyle \frac{\beta }{k}}+1}{B}_{{\displaystyle \frac{1}{2}}}\left(\mathfrak{n}-\frac{\beta }{k}+1,s+1\right)+{\displaystyle \frac{k}{2^s\left((\mathfrak{n}+s+1)k-\beta \right)}}\right)\left(\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_1,·)\right|+\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_2,·)\right|\right)(a.e.).\hfill \end{array}$$

Moreover, for $s=1$, we get the following midpoint inequality via stochastic convexity,

$$\begin{array}{cc}\hfill & \left|{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left(\frac{{\mathfrak{e}}_1+{\mathfrak{e}}_2}{2},·\right)-{\displaystyle \frac{2^{\mathfrak{n}-\frac{\beta }{k}-1}k{\mathsf{\Gamma }}_k\left(\mathfrak{n}-\frac{\beta }{k}+k\right)}{{\left({\mathfrak{e}}_2-{\mathfrak{e}}_1\right)}^{\mathfrak{n}-\frac{\beta }{k}}}}{\mathcal{C}}^{\beta ,k}\left[\mathcal{P}\right]\right|\hfill \\ \hfill \le & {\displaystyle \frac{k\left({\mathfrak{e}}_2-{\mathfrak{e}}_1\right)}{4\left(k\left(\mathfrak{n}+1\right)-\beta \right)}}\left(\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_1,·)\right|+\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_2,·)\right|\right)(a.e.),\hfill \end{array}$$

which was provided by Alruwaily et al. in ([28], Theorem 8).

Corollary 5.

If we set $\lambda =0$, Theorem 2 yields the following midpoint-type inequalities via stochastic s-convexity:

$$\begin{array}{cc}\hfill & \left|{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left(\frac{{\mathfrak{e}}_1+{\mathfrak{e}}_2}{2},·\right)-{\displaystyle \frac{2^{\mathfrak{n}-\frac{\beta }{k}-1}k{\mathsf{\Gamma }}_k\left(\mathfrak{n}-\frac{\beta }{k}+k\right)}{{\left({\mathfrak{e}}_2-{\mathfrak{e}}_1\right)}^{\mathfrak{n}-\frac{\beta }{k}}}}{\mathcal{C}}^{\beta ,k}\left[\mathcal{P}\right]\right|\hfill \\ \hfill \le & {\displaystyle \frac{{\mathfrak{e}}_2-{\mathfrak{e}}_1}{4}}{\left({\displaystyle \frac{k}{(k\mathfrak{n}-\beta )(p+1)}}{}_2{}^{ }F_1^{ }\left({\displaystyle \frac{k(\mathfrak{n}-1)-\beta }{k\mathfrak{n}-\beta }},1,p+2;1\right)\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & \times \left({\left({\displaystyle \frac{\left(2^{s+1}-1\right){\left|{\mathcal{P}}^{\prime }({\mathfrak{e}}_1,·)\right|}^q+{\left|{\mathcal{P}}^{\prime }({\mathfrak{e}}_2,·)\right|}^q}{2^s\left(s+1\right)}}\right)}^{{\displaystyle \frac{1}{q}}}+{\left({\displaystyle \frac{{\left|{\mathcal{P}}^{\prime }({\mathfrak{e}}_1,·)\right|}^q+\left(2^{s+1}-1\right){\left|{\mathcal{P}}^{\prime }({\mathfrak{e}}_2,·)\right|}^q}{2^s\left(s+1\right)}}\right)}^{{\displaystyle \frac{1}{q}}}\right)(a.e.).\hfill \end{array}$$

5.2. Trapezium-Type Inequalities

Corollary 6.

If we attempt to make $\lambda =1$, then Theorem 1 yields the following trapezium inequality via stochastic s-convexity:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left({\mathfrak{e}}_1,·\right)+{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left({\mathfrak{e}}_2,·\right)}{2}}-{\displaystyle \frac{2^{\mathfrak{n}-\frac{\beta }{k}-1}k{\mathsf{\Gamma }}_k\left(\mathfrak{n}-\frac{\beta }{k}+k\right)}{{\left({\mathfrak{e}}_2-{\mathfrak{e}}_1\right)}^{\mathfrak{n}-\frac{\beta }{k}}}}{\mathcal{C}}^{\beta ,k}\left[\mathcal{P}\right]\right|\hfill \\ \hfill \le & {\displaystyle \frac{{\mathfrak{e}}_2-{\mathfrak{e}}_1}{4}}\left({\displaystyle \frac{2-2^{-s}}{s+1}}-2^{\mathfrak{n}-{\displaystyle \frac{\beta }{k}}+1}{B}_{{\displaystyle \frac{1}{2}}}\left(\mathfrak{n}-\frac{\beta }{k}+1,s+1\right)+{\displaystyle \frac{k\mathfrak{n}-\beta }{2^s\left((\mathfrak{n}+s+1)k-\beta \right)(s+1)}}\right)\left(\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_1,·)\right|+\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_2,·)\right|\right)(a.e.).\hfill \end{array}$$

Moreover, for $s=1$, we get the following trapezium inequality via stochastic convexity,

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left({\mathfrak{e}}_1,·\right)+{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left({\mathfrak{e}}_2,·\right)}{2}}-{\displaystyle \frac{2^{\mathfrak{n}-\frac{\beta }{k}-1}k{\mathsf{\Gamma }}_k\left(\mathfrak{n}-\frac{\beta }{k}+k\right)}{{\left({\mathfrak{e}}_2-{\mathfrak{e}}_1\right)}^{\mathfrak{n}-\frac{\beta }{k}}}}{\mathcal{C}}^{\beta ,k}\left[\mathcal{P}\right]\right|\hfill \\ \hfill \le & {\displaystyle \frac{\left(k\mathfrak{n}-\beta \right)\left({\mathfrak{e}}_2-{\mathfrak{e}}_1\right)}{4\left(k\left(\mathfrak{n}+1\right)-\beta \right)}}\left(\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_1,·)\right|+\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_2,·)\right|\right)(a.e.),\hfill \end{array}$$

which was provided by Alruwaily et al. in [28] ([Theorem 11]).

Corollary 7.

If we set $\lambda =1$, Theorem 2 yields the following trapezium-type inequalities via stochastic s-convexity:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left({\mathfrak{e}}_1,·\right)+{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left({\mathfrak{e}}_2,·\right)}{2}}-{\displaystyle \frac{2^{\mathfrak{n}-\frac{\beta }{k}-1}k{\mathsf{\Gamma }}_k\left(\mathfrak{n}-\frac{\beta }{k}+k\right)}{{\left({\mathfrak{e}}_2-{\mathfrak{e}}_1\right)}^{\mathfrak{n}-\frac{\beta }{k}}}}{\mathcal{C}}^{\beta ,k}\left[\mathcal{P}\right]\right|\hfill \\ \hfill \le & {\displaystyle \frac{{\mathfrak{e}}_2-{\mathfrak{e}}_1}{4}}{\left({\displaystyle \frac{k}{k\mathfrak{n}-\beta }}B\left({\displaystyle \frac{k}{k\mathfrak{n}-\beta }},p+1\right)\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & \times \left({\left({\displaystyle \frac{\left(2^{s+1}-1\right){\left|{\mathcal{P}}^{\prime }({\mathfrak{e}}_1,·)\right|}^q+{\left|{\mathcal{P}}^{\prime }({\mathfrak{e}}_2,·)\right|}^q}{2^s\left(s+1\right)}}\right)}^{{\displaystyle \frac{1}{q}}}+{\left({\displaystyle \frac{{\left|{\mathcal{P}}^{\prime }({\mathfrak{e}}_1,·)\right|}^q+\left(2^{s+1}-1\right){\left|{\mathcal{P}}^{\prime }({\mathfrak{e}}_2,·)\right|}^q}{2^s\left(s+1\right)}}\right)}^{{\displaystyle \frac{1}{q}}}\right)(a.e.).\hfill \end{array}$$

5.3. Simpson-Type Inequalities

Corollary 8.

If we set $\lambda ={\displaystyle \frac{1}{3}}$, Theorem 1 yields the following Simpson-type inequalities via s-convexity:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left({\mathfrak{e}}_1,·\right)+4{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left(\frac{{\mathfrak{e}}_1+{\mathfrak{e}}_2}{2},·\right)+{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left({\mathfrak{e}}_2,·\right)}{6}}-{\displaystyle \frac{2^{\mathfrak{n}-\frac{\beta }{k}-1}k{\mathsf{\Gamma }}_k\left(\mathfrak{n}-\frac{\beta }{k}+k\right)}{{\left({\mathfrak{e}}_2-{\mathfrak{e}}_1\right)}^{\mathfrak{n}-\frac{\beta }{k}}}}{\mathcal{C}}^{\beta ,k}\left[\mathcal{P}\right]\right|\hfill \\ \hfill \le & {\displaystyle \frac{{\mathfrak{e}}_2-{\mathfrak{e}}_1}{4}}\left({\displaystyle \frac{2+2^{-s}-2^{1-s}{\left(2-2^{-\frac{k}{k\mathfrak{n}-\beta }}\right)}^{s+1}}{2(s+1)}}+2^{\mathfrak{n}-{\displaystyle \frac{\beta }{k}}+1}{B}_{{\displaystyle \frac{1}{2}}}\left(\mathfrak{n}-\frac{\beta }{k}+1,s+1\right)-2^{\mathfrak{n}-{\displaystyle \frac{\beta }{k}}+2}{B}_{{\displaystyle \frac{1}{2^{1+\frac{k}{k\mathfrak{n}-\beta }}}}}\left(\mathfrak{n}-\frac{\beta }{k}+1,s+1\right)\right.\hfill \\ \hfill & \left.+{\displaystyle \frac{2(k\mathfrak{n}-\beta )·2^{-\frac{(\mathfrak{n}+s+1)k-\beta }{k\mathfrak{n}-\beta }}+k(s+1)-\frac{1}{2}\left((\mathfrak{n}+s+1)k-\beta \right)}{2^s\left((\mathfrak{n}+s+1)k-\beta \right)(s+1)}}\right)\left(\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_1,·)\right|+\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_2,·)\right|\right)(a.e.).\hfill \end{array}$$

Moreover, by setting $s=1$, we get

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left({\mathfrak{e}}_1,·\right)+4{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left(\frac{{\mathfrak{e}}_1+{\mathfrak{e}}_2}{2},·\right)+{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left({\mathfrak{e}}_2,·\right)}{6}}-{\displaystyle \frac{2^{\mathfrak{n}-\frac{\beta }{k}-1}k{\mathsf{\Gamma }}_k\left(\mathfrak{n}-\frac{\beta }{k}+k\right)}{{\left({\mathfrak{e}}_2-{\mathfrak{e}}_1\right)}^{\mathfrak{n}-\frac{\beta }{k}}}}{\mathcal{C}}^{\beta ,k}\left[\mathcal{P}\right]\right|\hfill \\ \hfill \le & {\displaystyle \frac{{\mathfrak{e}}_2-{\mathfrak{e}}_1}{4}}\left({\displaystyle \frac{2\left(k\mathfrak{n}-\beta \right)-3^{\frac{k}{\mathfrak{n}k-\beta }}\left(k\left(\mathfrak{n}+1\right)-\beta \right)+3^{\frac{k\left(\mathfrak{n}+1\right)-\beta }{\mathfrak{n}k-\beta }}k}{3^{\frac{k\left(\mathfrak{n}+1\right)-\beta }{\mathfrak{n}k-\beta }}\left(k\left(\mathfrak{n}+1\right)-\beta \right)}}\right)\left(\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_1,·)\right|+\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_2,·)\right|\right)(a.e.).\hfill \end{array}$$

Corollary 9.

If we set $\lambda ={\displaystyle \frac{1}{3}}$, Theorem 2 yields the following Simpson-type inequalities via stochastic s-convexity:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left({\mathfrak{e}}_1,·\right)+4{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left(\frac{{\mathfrak{e}}_1+{\mathfrak{e}}_2}{2},·\right)+{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left({\mathfrak{e}}_2,·\right)}{6}}-{\displaystyle \frac{2^{\mathfrak{n}-\frac{\beta }{k}-1}k{\mathsf{\Gamma }}_k\left(\mathfrak{n}-\frac{\beta }{k}+k\right)}{{\left({\mathfrak{e}}_2-{\mathfrak{e}}_1\right)}^{\mathfrak{n}-\frac{\beta }{k}}}}{\mathcal{C}}^{\beta ,k}\left[\mathcal{P}\right]\right|\hfill \\ \hfill \le & {\displaystyle \frac{{\mathfrak{e}}_2-{\mathfrak{e}}_1}{4}}{\left({\displaystyle \frac{k}{k\mathfrak{n}-\beta }}{\left({\displaystyle \frac{1}{2}}\right)}^{p+{\displaystyle \frac{k}{k\mathfrak{n}-\beta }}}B\left({\displaystyle \frac{k}{k\mathfrak{n}-\beta }},p+1\right)+{\displaystyle \frac{k}{(k\mathfrak{n}-\beta )(p+1)}}{\left({\displaystyle \frac{1}{2}}\right)}^{p+1}{}_2{}^{ }F_1^{ }\left({\displaystyle \frac{k(\mathfrak{n}-1)-\beta }{k\mathfrak{n}-\beta }},1,p+2;{\displaystyle \frac{1}{2}}\right)\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & \times \left({\left({\displaystyle \frac{\left(2^{s+1}-1\right){\left|{\mathcal{P}}^{\prime }({\mathfrak{e}}_1,·)\right|}^q+{\left|{\mathcal{P}}^{\prime }({\mathfrak{e}}_2,·)\right|}^q}{2^s\left(s+1\right)}}\right)}^{{\displaystyle \frac{1}{q}}}+{\left({\displaystyle \frac{{\left|{\mathcal{P}}^{\prime }({\mathfrak{e}}_1,·)\right|}^q+\left(2^{s+1}-1\right){\left|{\mathcal{P}}^{\prime }({\mathfrak{e}}_2,·)\right|}^q}{2^s\left(s+1\right)}}\right)}^{{\displaystyle \frac{1}{q}}}\right)(a.e.).\hfill \end{array}$$

5.4. Bullen-Type Inequalities

Corollary 10.

If we set $\lambda ={\displaystyle \frac{1}{2}}$, Theorem 1 yields the following Bullen-type inequalities via stochastic s-convexity:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left({\mathfrak{e}}_1,·\right)+2{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left(\frac{{\mathfrak{e}}_1+{\mathfrak{e}}_2}{2},·\right)+{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left({\mathfrak{e}}_2,·\right)}{4}}-{\displaystyle \frac{2^{\mathfrak{n}-\frac{\beta }{k}-1}k{\mathsf{\Gamma }}_k\left(\mathfrak{n}-\frac{\beta }{k}+k\right)}{{\left({\mathfrak{e}}_2-{\mathfrak{e}}_1\right)}^{\mathfrak{n}-\frac{\beta }{k}}}}{\mathcal{C}}^{\beta ,k}\left[\mathcal{P}\right]\right|\hfill \\ \hfill \le & {\displaystyle \frac{{\mathfrak{e}}_2-{\mathfrak{e}}_1}{4}}\left({\displaystyle \frac{2+2^{-s}-2^{1-s}{\left(2-3^{-\frac{k}{k\mathfrak{n}-\beta }}\right)}^{s+1}}{3(s+1)}}+2^{\mathfrak{n}-{\displaystyle \frac{\beta }{k}}+1}{B}_{{\displaystyle \frac{1}{2}}}\left(\mathfrak{n}-\frac{\beta }{k}+1,s+1\right)-2^{\mathfrak{n}-{\displaystyle \frac{\beta }{k}}+2}{B}_{{\displaystyle \frac{1}{2}}·3^{-{\displaystyle \frac{k}{k\mathfrak{n}-\beta }}}}\left(\mathfrak{n}-\frac{\beta }{k}+1,s+1\right)\right.\hfill \\ \hfill & \left.+{\displaystyle \frac{2(k\mathfrak{n}-\beta )·3^{-\frac{(\mathfrak{n}+s+1)k-\beta }{k\mathfrak{n}-\beta }}+k(s+1)-\frac{1}{3}\left((\mathfrak{n}+s+1)k-\beta \right)}{2^s\left((\mathfrak{n}+s+1)k-\beta \right)(s+1)}}\right)\left(\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_1,·)\right|+\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_2,·)\right|\right)(a.e.).\hfill \end{array}$$

Moreover, for $s=1$ we get the following Bullen inequality via stochastic convexity

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left({\mathfrak{e}}_1,·\right)+2{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left(\frac{{\mathfrak{e}}_1+{\mathfrak{e}}_2}{2},·\right)+{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left({\mathfrak{e}}_2,·\right)}{4}}-{\displaystyle \frac{2^{\mathfrak{n}-\frac{\beta }{k}-1}k{\mathsf{\Gamma }}_k\left(\mathfrak{n}-\frac{\beta }{k}+k\right)}{{\left({\mathfrak{e}}_2-{\mathfrak{e}}_1\right)}^{\mathfrak{n}-\frac{\beta }{k}}}}{\mathcal{C}}^{\beta ,k}\left[\mathcal{P}\right]\right|\hfill \\ \hfill \le & {\displaystyle \frac{{\mathfrak{e}}_2-{\mathfrak{e}}_1}{4}}\left({\displaystyle \frac{2\left(k\mathfrak{n}-\beta \right)-2^{\frac{k}{\mathfrak{n}k-\beta }}\left(k\left(\mathfrak{n}+1\right)-\beta \right)+2^{\frac{k\left(\mathfrak{n}+1\right)-\beta }{\mathfrak{n}k-\beta }}k}{2^{\frac{k\left(\mathfrak{n}+1\right)-\beta }{\mathfrak{n}k-\beta }}\left(k\left(\mathfrak{n}+1\right)-\beta \right)}}\right)\left(\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_1,·)\right|+\left|{\mathcal{P}}^{\left(\mathfrak{n}+1\right)}({\mathfrak{e}}_2,·)\right|\right)(a.e.).\hfill \end{array}$$

Corollary 11.

If we set $\lambda ={\displaystyle \frac{1}{2}}$, Theorem 2 yields the following Bullen-type inequalities via stochastic s-convexity:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left({\mathfrak{e}}_1,·\right)+2{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left(\frac{{\mathfrak{e}}_1+{\mathfrak{e}}_2}{2},·\right)+{\mathcal{P}}^{\left(\mathfrak{n}\right)}\left({\mathfrak{e}}_2,·\right)}{4}}-{\displaystyle \frac{2^{\mathfrak{n}-\frac{\beta }{k}-1}k{\mathsf{\Gamma }}_k\left(\mathfrak{n}-\frac{\beta }{k}+k\right)}{{\left({\mathfrak{e}}_2-{\mathfrak{e}}_1\right)}^{\mathfrak{n}-\frac{\beta }{k}}}}{\mathcal{C}}^{\beta ,k}\left[\mathcal{P}\right]\right|\hfill \\ \hfill \le & {\displaystyle \frac{{\mathfrak{e}}_2-{\mathfrak{e}}_1}{4}}{\left({\displaystyle \frac{k}{k\mathfrak{n}-\beta }}{\left({\displaystyle \frac{1}{3}}\right)}^{p+{\displaystyle \frac{k}{k\mathfrak{n}-\beta }}}B\left({\displaystyle \frac{k}{k\mathfrak{n}-\beta }},p+1\right)+{\displaystyle \frac{k}{(k\mathfrak{n}-\beta )(p+1)}}{\left({\displaystyle \frac{2}{3}}\right)}^{p+1}{}_2{}^{ }F_1^{ }\left({\displaystyle \frac{k(\mathfrak{n}-1)-\beta }{k\mathfrak{n}-\beta }},1,p+2;{\displaystyle \frac{2}{3}}\right)\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & \times \left({\left({\displaystyle \frac{\left(2^{s+1}-1\right){\left|{\mathcal{P}}^{\prime }({\mathfrak{e}}_1,·)\right|}^q+{\left|{\mathcal{P}}^{\prime }({\mathfrak{e}}_2,·)\right|}^q}{2^s\left(s+1\right)}}\right)}^{{\displaystyle \frac{1}{q}}}+{\left({\displaystyle \frac{{\left|{\mathcal{P}}^{\prime }({\mathfrak{e}}_1,·)\right|}^q+\left(2^{s+1}-1\right){\left|{\mathcal{P}}^{\prime }({\mathfrak{e}}_2,·)\right|}^q}{2^s\left(s+1\right)}}\right)}^{{\displaystyle \frac{1}{q}}}\right)(a.e.).\hfill \end{array}$$

6. A Numerical Case Study Supporting the Theoretical Findings

This section presents a numerical example with graphical illustrations that corroborate the validity of the results established in this study.

Example 1.

Consider the stochastic process $\mathcal{P}$ on the interval $\mathcal{I}=[0,1]$ given by $\mathcal{P}(\mathfrak{u},·)={\displaystyle \frac{{\mathfrak{u}}^{\mathfrak{n}+2}}{(\mathfrak{n}+2)!}}$. Its $(\mathfrak{n}+1)$–th derivative, ${\mathcal{P}}^{(\mathfrak{n}+1)}\left(\mathfrak{u}\right)=\mathfrak{u}$, is convex over $\mathcal{I}$. Under these conditions, Corollary 1 leads to the following parametrized inequalities:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1+\lambda }{8}}-2^{\mathfrak{n}-{\displaystyle \frac{\beta }{k}}-2}{\displaystyle \frac{k\mathfrak{n}-\beta }{k}}\left(B_{{\displaystyle \frac{1}{2}}}\left(\mathfrak{n}-{\displaystyle \frac{\beta }{k}},3\right)+{\left({\displaystyle \frac{1}{2}}\right)}^{\mathfrak{n}-{\displaystyle \frac{\beta }{k}}+2}{\displaystyle \frac{k}{k(\mathfrak{n}+2)-\beta }}\right)\right|\hfill \\ \hfill \le & {\displaystyle \frac{2\left(k\mathfrak{n}-\beta \right){\lambda }^{\frac{k\left(\mathfrak{n}+1\right)-\beta }{\mathfrak{n}k-\beta }}-\left(k\left(\mathfrak{n}+1\right)-\beta \right)\lambda +k}{4\left(k\left(\mathfrak{n}+1\right)-\beta \right)}}.\hfill \end{array}$$

(10)

The above result depends on three parameters. Both sides of Inequality (10) are displayed in Figure 1 as functions of $\beta \in (0,5]$ and $\lambda \in [0,1]$ for several fixed values of the parameter k. These visualizations confirm that the left-hand side is consistently bounded above by the right-hand side across a range of parameter values, thereby validating the correctness of our theoretical result.

7. Conclusions

In this study, we successfully constructed a versatile parametric identity for an $(\mathfrak{n}+1)$-times mean-square-differentiable process involving a stochastic k-Caputo fractional derivative. This identity enables the systematic derivation of a wide spectrum of fractional inequalities under stochastic uncertainty for s-convex stochastic processes. Importantly, the proposed framework is sufficiently general to encapsulate classical deterministic inequalities, such as midpoint, trapezium, Simpson, and Bullen, as limiting or special cases through appropriate parameter selection. Directions for future research include extending this framework to coordinate convex stochastic processes and developing numerical schemes for the associated stochastic fractional differential equations.