On Fractional Hermite–Hadamard-Type Inequalities for Harmonically s-Convex Stochastic Processes

Abstract

In this paper, we investigate Hermite–Hadamard-type inequalities for harmonically s-convex stochastic processes via Riemann–Liouville fractional integrals. We begin by introducing the notion of harmonically s-convex stochastic processes. Subsequently, we establish a variety of Riemann–Liouville fractional Hermite–Hadamard-type inequalities for harmonic s-convex stochastic. We first provide the Hermite–Hadamard inequality, then by introducing a novel identity involving mean-square stochastic Riemann–Liouville fractional integral operators, we derive several midpoint-type inequalities for harmonically s-convex stochastic processes. Illustrative example with graphical depiction and a practical application are provided.

1. Introduction

Convexity is one of the most fundamental and powerful concepts in mathematical analysis, optimization, and applied mathematics. A real-valued function $\mathrm{g}:I\subseteq \mathbb{R}\to \mathbb{R}$ defined on an interval I is said to be convex if, for all $u,v\in I$ and $\nu \in [0,1]$, the inequality

$$\mathrm{g}(\nu u+(1-\nu \left)v\right)\le \nu \mathrm{g}\left(u\right)+(1-\nu )\mathrm{g}\left(v\right)$$

holds. This elementary definition captures the intuitive notion that the graph of a convex function lies below the chord joining any two points on the graph. Convex functions enjoy numerous desirable properties making them indispensable in variational analysis, economics, engineering, and numerical methods.

Among the most celebrated results in the theory of convex functions is the Hermite–Hadamard inequality, which provides a double-sided estimate for the integral average of a convex function over an interval. Precisely, if $\mathrm{g}:[\varphi ,\psi ]\to \mathbb{R}$ is convex, then

$$\mathrm{g}\left({\displaystyle \frac{\varphi +\psi }{2}}\right)\le {\displaystyle \frac{1}{\psi -\varphi }}\underset{\varphi }{\stackrel{\psi }{\int }}\mathrm{g}\left(u\right)du\le {\displaystyle \frac{\mathrm{g}\left(\varphi \right)+\mathrm{g}\left(\psi \right)}{2}}.$$

This elegant inequality not only characterizes convexity (in the sense that a continuous function satisfying the above for all subintervals is convex) but also serves as a cornerstone for deriving numerous integral inequalities in mathematical analysis. Over the past century, the Hermite–Hadamard inequality has inspired a vast body of research aimed at refining, extending, and generalizing it under various assumptions on the function class and the integration operator [1,2,3,4].

A particularly active line of investigation concerns midpoint-type inequalities, which refine the left-hand side of the Hermite–Hadamard inequality by providing tighter bounds or error estimates for the difference between the midpoint value $\mathrm{g}\left({\displaystyle \frac{\varphi +\psi }{2}}\right)$ and the integral mean ${\displaystyle \frac{1}{\psi -\varphi }}\underset{\varphi }{\stackrel{\psi }{\int }}\mathrm{g}\left(u\right)du$. Such inequalities are crucial in numerical integration, especially in estimating the error of the midpoint rule, and have been extensively studied for functions with additional smoothness or convexity properties (e.g., Lipschitz continuity, bounded derivatives, or higher-order convexity).

In [5], Kırmacı proved the following midpoint-type inequality.

Theorem 1.

Let $\mathrm{g}$ be a differentiable function satisfying $\mathrm{g}\in L\left([\varphi ,\psi ]\right)$ with $\varphi <\psi$. If $\left|{\mathrm{g}}^{\prime }\right|$ is convex, then

$$\left|\mathrm{g}\left({\displaystyle \frac{\varphi +\psi }{2}}\right)-{\displaystyle \frac{1}{\psi -\varphi }}\underset{\varphi }{\stackrel{\psi }{\int }}\mathrm{g}\left(u\right)du\right|\le {\displaystyle \frac{\psi -\varphi }{8}}\left(\left|{\mathrm{g}}^{\prime }\left(\varphi \right)\right|+\left|{\mathrm{g}}^{\prime }\left(\psi \right)\right|\right).$$

In parallel with these developments, researchers have recognized that the classical notion of convexity is often too restrictive for modeling real-world phenomena or capturing the behavior of functions arising in applied contexts. This has led to the introduction of numerous generalized convexity concepts, each relaxing or modifying the standard definition to accommodate broader classes of functions while preserving some of the analytical tractability of convexity. Notable examples include s-convexity, quasi-convexity, log-convexity, P-convexity, and harmonic convexity.

In [6], Hudzik and Maligranda considered the class of s-convex functions in the second sense. This class is defined for functions with non-negative arguments as follows: For a fixed $s\in (0,1]$ if for all $u,v\in I$ and $\nu \in [0,1]$ the inequality

$$\mathrm{g}(\nu u+(1-\nu )v)\le {\nu }^s\mathrm{g}\left(u\right)+{(1-\nu )}^s\mathrm{g}\left(v\right)$$

holds, then $\mathrm{g}$ is said to be s-convex in the second sense on I.

Dragomir and Fitzpatrick presented the Hermite–Hadamard inequality for s-convex functions in [7] as follows: If, for a fixed $s\in (0,1]$, the function $\mathrm{g}:[\varphi ,\psi ]\subset (0,\infty )\to \mathbb{R}$ is s-convex in the second sense, then

$$2^{s-1}\mathrm{g}\left({\displaystyle \frac{\varphi +\psi }{2}}\right)\le {\displaystyle \frac{1}{\psi -\varphi }}\underset{\varphi }{\stackrel{\psi }{\int }}\mathrm{g}\left(u\right)du\le {\displaystyle \frac{\mathrm{g}\left(\varphi \right)+\mathrm{g}\left(\psi \right)}{s+1}}.$$

The midpoint-type inequality for differentiable s-convex functions was provided by Liu in [8] in the following manner:

Theorem 2.

Let $\mathrm{g}$ be a differentiable function satisfying $\mathrm{g}\in L\left([\varphi ,\psi ]\right)$ with $\varphi <\psi$. If $\left|{\mathrm{g}}^{\prime }\right|$ is s-convex in the second sense for a fixed $s\in (0,1]$, then

$$\left|\mathrm{g}\left({\displaystyle \frac{\varphi +\psi }{2}}\right)-{\displaystyle \frac{1}{\psi -\varphi }}\underset{\varphi }{\stackrel{\psi }{\int }}\mathrm{g}\left(u\right)du\right|\le {\displaystyle \frac{(2^{s+1}-1)(\psi -\varphi )}{2^{s+1}(s+1)(s+2)}}\left(\left|{\mathrm{g}}^{\prime }\left(\varphi \right)\right|+\left|{\mathrm{g}}^{\prime }\left(\psi \right)\right|\right).$$

Harmonically convex functions were first systematically studied by İşcan in [9] as a natural counterpart to classical convexity when the argument of the function is transformed via the reciprocal map. A function $\mathrm{g}:I\subseteq (0,\infty )\to \mathbb{R}$ is said to be harmonically convex if for all $u,v\in I$ and $\nu \in [0,1]$,

$$\mathrm{g}\left({\displaystyle \frac{uv}{(1-\nu )u+\nu v}}\right)\le \nu \mathrm{g}\left(u\right)+(1-\nu )\mathrm{g}\left(v\right).$$

In the same paper, the author established the Hermite–Hadamard inequality for harmonically convex functions: if $\mathrm{g}:[\varphi ,\psi ]\subseteq (0,\infty )\to \mathbb{R}$ is harmonically convex, then

$$\mathrm{g}\left({\displaystyle \frac{2\varphi \psi }{\varphi +\psi }}\right)\le {\displaystyle \frac{\varphi \psi }{\psi -\varphi }}\underset{\varphi }{\stackrel{\psi }{\int }}{\displaystyle \frac{\mathrm{g}\left(u\right)}{u^2}}du\le {\displaystyle \frac{\mathrm{g}\left(\varphi \right)+\mathrm{g}\left(\psi \right)}{2}}.$$

Here, the harmonic mean ${\displaystyle \frac{2\varphi \psi }{\varphi +\psi }}$ naturally replaces the arithmetic mean ${\displaystyle \frac{\varphi +\psi }{2}}$, reflecting the reciprocal structure inherent in harmonic convexity.

In a separate work [10], the same author proved the following midpoint-type inequality for differentiable harmonically convex functions:

Theorem 3.

Let $\mathrm{g}$ be a differentiable function satisfying $\mathrm{g}\in L\left([\varphi ,\psi ]\right)$ with $\varphi <\psi$. If $\left|{\mathrm{g}}^{\prime }\right|$ is harmonically convex on $[\varphi ,\psi ]$, then

$$\left|\mathrm{g}\left({\displaystyle \frac{2\varphi \psi }{\varphi +\psi }}\right)-{\displaystyle \frac{\varphi \psi }{\psi -\varphi }}\underset{\varphi }{\stackrel{\psi }{\int }}{\displaystyle \frac{\mathrm{g}\left(u\right)}{u^2}}du\right|\le {\displaystyle \frac{\varphi \psi }{2}}\left[{\displaystyle \frac{2}{\psi -\varphi }}ln\left({\displaystyle \frac{2\varphi }{\varphi +\psi }}\right)-{\displaystyle \frac{2}{\varphi +\psi }}\right]\left(\left|{\mathrm{g}}^{\prime }\left(\varphi \right)\right|+\left|{\mathrm{g}}^{\prime }\left(\psi \right)\right|\right).$$

Further generalizations have emerged by combining harmonic convexity with other convexity parameters. Of particular relevance to the present work is the notion of harmonically s-convex functions, introduced in [11] as a hybrid of harmonic convexity and s-convexity. For a fixed $s\in (0,1]$, a function $\mathrm{g}:I\subseteq (0,\infty )\to \mathbb{R}$ is considered harmonically s-convex if the following inequality holds:

$$\mathrm{g}\left({\displaystyle \frac{uv}{(1-\nu )u+\nu v}}\right)\le {\nu }^s\mathrm{g}\left(u\right)+{(1-\nu )}^s\mathrm{g}\left(v\right),\forall u,v\in I,\nu \in [0,1].$$

When $s=1$, this reduces to ordinary harmonic convexity. This concept has proven fruitful in deriving new integral inequalities, especially when combined with fractional calculus.

In the same paper, the author established the Hermite–Hadamard inequality for harmonically s-convex functions: if $\mathrm{g}:[\varphi ,\psi ]\subseteq (0,\infty )\to \mathbb{R}$ is harmonically s-convex, then

$$2^{s-1}\mathrm{g}\left({\displaystyle \frac{2\varphi \psi }{\varphi +\psi }}\right)\le {\displaystyle \frac{\varphi \psi }{\psi -\varphi }}\underset{\varphi }{\stackrel{\psi }{\int }}{\displaystyle \frac{\mathrm{g}\left(u\right)}{u^2}}du\le {\displaystyle \frac{\mathrm{g}\left(\varphi \right)+\mathrm{g}\left(\psi \right)}{s+1}}.$$

Indeed, the intersection of generalized convexity and fractional calculus has become a vibrant area of research in recent years. Fractional integrals, such as the Riemann–Liouville, Hadamard, Katugampola, and conformable fractional integrals, offer refined tools for modeling memory effects, non-local phenomena, and anomalous diffusion. By replacing the classical integral in the Hermite–Hadamard inequality with a fractional integral operator, researchers have obtained fractional Hermite–Hadamard-type inequalities for various classes of convex functions, including harmonically convex and harmonically s-convex functions (see, e.g., [12,13,14,15,16]).

In [17], İşcan and Wu established the following Hermite–Hadamard inequalities for harmonically convex functions via fractional integrals: If for a fixed $s\in (0,1]$, the function $\mathrm{g}:[\varphi ,\psi ]\subset [0,\infty )\to \mathbb{R}$ is harmonically s-convex, then

$$\mathrm{g}\left({\displaystyle \frac{2\varphi \psi }{\varphi +\psi }}\right)\le {\displaystyle \frac{\mathrm{\Gamma }\left(\alpha +1\right)}{2}}{\left({\displaystyle \frac{\varphi \psi }{\psi -\varphi }}\right)}^{\alpha }\left({\mathbf{J}}_{{\frac{1}{\varphi }}^{-}}^{\alpha }\left(\mathrm{g}\circ \mathrm{h}\right)\left({\displaystyle \frac{1}{\psi }}\right)+{\mathbf{J}}_{{\frac{1}{\psi }}^{+}}^{\alpha }\left(\mathrm{g}\circ \mathrm{h}\right)\left({\displaystyle \frac{1}{\varphi }}\right)\right)\le {\displaystyle \frac{\mathrm{g}\left(\varphi \right)+\mathrm{g}\left(\psi \right)}{2}},$$

where $\mathrm{h}\left(u\right)={\displaystyle \frac{1}{u}}$, and ${\mathbf{J}}_{{\sigma }^{+}}^{\alpha }$ and ${\mathbf{J}}_{{\sigma }^{-}}^{\alpha }$ denote the left and right Riemann–Liouville fractional integrals [18].

In modern mathematical analysis, stochastic processes offer a natural framework for studying families of random variables indexed by a real parameter. Formally, a stochastic process is a collection of random variables $\left\{\mathbb{X}\right(v):v\in I\}$, where $I\subseteq \mathbb{R}$ is an interval, all defined on a common probability space $(\mathrm{\Omega },\mathcal{J},\mathbb{P})$. Such objects arise in various areas including mathematical statistics, optimization, and applied analysis, whenever one wishes to incorporate randomness into functional relationships. When analytical properties such as convexity are extended to this probabilistic setting, it becomes natural to consider convex stochastic processes and their generalizations.

The notion of convexity for stochastic processes was first formalized by Nikodem in [19]. We recall it as follows.

Definition 1

([19]). A stochastic process $\mathbb{X}:\mathcal{D}\times \mathrm{\Lambda }\to \mathbb{R}$ is said to be convex if, for any two points $u,v\in \mathcal{D}$ and every $\nu \in [0,1]$, the inequality

$$\mathbb{X}\left(\nu u+(1-\nu )v,·\right)\le \nu \mathbb{X}(u,·)+(1-\nu )\mathbb{X}(v,·)(\mathit{almost}\mathit{everywhere}).$$

Building on this concept, Kotrys established in [20] the stochastic counterpart of the classical Hermite–Hadamard inequality:

$$\mathbb{X}\left({\displaystyle \frac{\varphi +\psi }{2}},·\right)\le {\displaystyle \frac{1}{\psi -\varphi }}\underset{\varphi }{\stackrel{\psi }{\int }}\mathbb{X}(u,·)du\le {\displaystyle \frac{\mathbb{X}(\varphi ,·)+\mathbb{X}(\psi ,·)}{2}},$$

valid for any convex stochastic process $\mathbb{X}$ on an interval $[\varphi ,\psi ]\subset \mathcal{D}$.

A natural generalization, known as s-convex stochastic processes in the second sense, was introduced by Set et al. [21] in the stochastic setting.

Definition 2

([21]). Let $s\in (0,1]$. A stochastic process $\mathbb{X}:\mathcal{D}\times \mathrm{\Lambda }\to \mathbb{R}$, with $\mathcal{D}\subset (0,\infty )$, is called s-convex in the second sense if, for all $u,v\in \mathcal{D}$ and every $\nu \in [0,1]$, one has

$$\mathbb{X}\left(\nu u+(1-\nu )v,·\right)\le {\nu }^s\mathbb{X}(u,·)+{(1-\nu )}^s\mathbb{X}(v,·)(a.e.).$$

Under this assumption, the authors of [21] derived the following Hermite–Hadamard-type bounds:

$$2^{s-1}\mathbb{X}\left({\displaystyle \frac{\varphi +\psi }{2}},·\right)\le {\displaystyle \frac{1}{\psi -\varphi }}\underset{\varphi }{\stackrel{\psi }{\int }}\mathbb{X}(u,·)du\le {\displaystyle \frac{\mathbb{X}(\varphi ,·)+\mathbb{X}(\psi ,·)}{s+1}}.$$

In [22], Okur and co-authors introduced the notion of harmonically convex stochastic processes in the following manner:

Definition 3.

A stochastic process $\mathbb{X}:\mathcal{D}\times \mathrm{\Lambda }\to \mathbb{R}$, with $\mathcal{D}\subset \mathbb{R}\setminus \left\{0\right\}$, is deemed to be harmonically convex stochastic process, if for all $u,v\in \mathcal{D}$ and every $\nu \in [0,1]$, one has

$$\mathbb{X}\left({\displaystyle \frac{uv}{(1-\nu )u+\nu v}},·\right)\le \nu \mathbb{X}(u,·)+(1-\nu )\mathbb{X}(v,·)(a.e.).$$

In the same paper, the authors established the Hermite–Hadamard inequality for harmonically convex stochastic processes: if $\mathbb{X}:[\varphi ,\psi ]\subset \mathbb{R}\setminus \left\{0\right\}\times \mathrm{\Lambda }\to \mathbb{R}$ is harmonically convex stochastic process, then

$$\mathbb{X}\left({\displaystyle \frac{2\varphi \psi }{\varphi +\psi }},·\right)\le {\displaystyle \frac{\varphi \psi }{\psi -\varphi }}\underset{\varphi }{\stackrel{\psi }{\int }}{\displaystyle \frac{\mathbb{X}(u,·)}{u^2}}du\le {\displaystyle \frac{\mathbb{X}(\varphi ,·)+\mathbb{X}(\psi ,·)}{2}}.$$

Agahi and Babakhani obtained the fractional version of Hermite–Hadamard inequality in [23] as follows:

$$\mathbb{X}\left({\displaystyle \frac{\varphi +\psi }{2}},·\right)\le {\displaystyle \frac{\mathrm{\Gamma }\left(\alpha +1\right)}{2{\left(\psi -\varphi \right)}^{\alpha }}}\left({\mathbf{J}}_{{\psi }^{-}}^{\alpha }\mathbb{X}\left(\varphi \right)+{\mathbf{J}}_{{\varphi }^{+}}^{\alpha }\mathbb{X}\left(\psi \right)\right)\le {\displaystyle \frac{\mathbb{X}\left(\varphi ,·\right)+\mathbb{X}\left(\psi ,·\right)}{2}}.$$

Recent studies have increasingly emphasized how generalized notions of convexity play a central role in the analysis of stochastic processes, particularly through the lens of integral inequalities. For instance, Materano et al. [24,25] established Simpson-type inequalities for stochastic processes that are s-convex or quasi-convex, and also provided Ostrowski-type estimates in the convex, s-convex, and quasi-convex settings. Deng and Wang [26] extended this line of inquiry by formulating fractional Hermite–Hadamard inequalities for functions exhibiting $(\beta ,m)$-logarithmic convexity, thus broadening the applicability of fractional calculus to stochastic frameworks. Afzal et al. [27] explored Jensen- and Hermite–Hadamard-type inequalities within the context of h-Godunova–Levin stochastic processes, highlighting the significance of such generalized convex structures. In [28], Meftah et al. derived Maclaurin-type inequalities for s-convex stochastic processes using stochastic k-Riemann–Liouville fractional integrals. Furthermore, Sharma et al. [29] introduced Hermite–Hadamard-type bounds for multidimensional h-harmonic preinvex stochastic processes. A growing body of literature continues to enrich both the theoretical underpinnings and real-world relevance of generalized convexity in stochastic analysis [30,31,32,33,34,35,36].

Motivated by these considerations, the present paper aims to introduce the notion of harmonically s-convex stochastic processes and establish related Riemann–Liouville fractional Hermite–Hadamard-type inequalities. The remainder of this paper is organized as follows: Section 2 recalls essential definitions and preliminary results concerning stochastic theory and fractional calculus. Section 3 presents some auxiliary results useful for the derivation of our main outcomes. Section 4 introduces the concept of harmonically s-convex stochastic processes, and present the stochastic Riemann-Liouville fractional Hermite–Hadamard inequality for that class. In Section 5, we begin by introducing a new fractional identity involving Riemann–Liouville fractional integrals, then use it to derive numerous fractional midpoint-type inequalities for mean-square differentiable harmonically s-convex stochastic processes. An illustrative example xith graphical representation as well as some practical applications are provided in Section 6. Section 7 concludes with remarks on potential extensions and applications.

2. Preliminaries

In this section, we collect the essential definitions and fundamental results from stochastic theory and fractional calculus as well as some special functions that will be instrumental throughout the paper. This section presents essential notions from stochastic theory and fractional calculus that will be used throughout the rest of this work.

Definition 4

([37]). Let $(\mathrm{\Omega },\mathit{A},\mathit{P})$ be a probability space. A function $\mathbb{X}:\mathrm{\Omega }\to \mathbb{R}$ is a random variable if it is $\mathit{A}$-measurable. A function $\mathbb{X}:\mathcal{D}\times \mathrm{\Lambda }\to \mathbb{R}$, where $\mathcal{D}\subseteq \mathbb{R}$ is an interval, is a stochastic process if for every $\nu \in \mathcal{D}$ the function $\mathbb{X}(\nu ,.)$ is a random variable.

Definition 5

([37]). A stochastic process $\mathbb{X}:\mathcal{D}\times \mathrm{\Lambda }\to \mathbb{R}$ is said to be continuous in the interval $\mathcal{D}$, if for all ${\nu }_0\in \mathcal{D}$ we have

$$\mathit{P}-\underset{\nu \to {\nu }_0}{lim}\mathbb{X}\left(\nu ,.\right)=\mathbb{X}\left({\nu }_0,.\right),$$

where $\mathit{P}-lim$ denotes the limit in probability.

Definition 6

([37]). $\mathbb{X}:\mathcal{D}\times \mathrm{\Lambda }\to \mathbb{R}$ is said to be mean-square continuous in $\mathcal{D}$, if for all ${\nu }_0\in \mathcal{D}$ we have

$$\underset{\nu \to {\nu }_0}{lim}\mathbb{E}\left[{\left(\mathbb{X}\left(\nu ,.\right)-\mathbb{X}\left({\nu }_0,.\right)\right)}^2\right]=0$$

where $\mathbb{E}\left[\mathbb{X}\left(\nu ,.\right)\right]$ denotes the expectation value of the random variable $\mathbb{X}\left(\nu ,.\right)$.

Definition 7

([37]). $\mathbb{X}:\mathcal{D}\times \mathrm{\Lambda }\to \mathbb{R}$ is said to be differentiable at a point $\nu \in \mathcal{D}$, if there is a random variable ${\mathbb{X}}^{\prime }:\mathcal{D}\times \mathrm{\Lambda }\to \mathbb{R}$

$${\mathbb{X}}^{\prime }\left(\nu ,.\right)=\mathit{P}-\underset{\nu \to {\nu }_0}{lim}{\displaystyle \frac{\mathbb{X}\left(\nu ,.\right)-\mathbb{X}\left({\nu }_0,.\right)}{\nu -{\nu }_0}}.$$

Definition 8

([37]). $\mathbb{X}:\mathcal{D}\times \mathrm{\Lambda }\to \mathbb{R}$ is said to be mean-square differentiable at a point $\nu \in \mathcal{D}$, if there is a random variable ${\mathbb{X}}^{\prime }:\mathcal{D}\times \mathrm{\Lambda }\to \mathbb{R}$

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

Definition 9

([37]). Let $\mathbb{X}:\mathcal{D}\times \mathrm{\Lambda }\to \mathbb{R}$ be a stochastic process with $E\left[\mathbb{X}{\left(\nu \right)}^2\right]\le \infty$, where $\nu \in \mathcal{D}$. Then, the random variable represented by $\mathbb{Y}:\mathrm{\Omega }\to \mathbb{R}$ is said to be a mean-square integral of the process $\mathbb{X}$ on $[\varphi ,\psi ]$ if for all sequences of partitions of the interval $[\varphi ,\psi ]\subseteq \mathcal{D}$, $\varphi ={\nu }_0<{\nu }_1<...<{\nu }_m=\psi$ and for all ${\mu }_k\in [{\nu }_{i-1},{\nu }_i]$, $i=1,...,n$ we have

$$\underset{\nu \to {\nu }_0}{lim}\mathbb{E}\left[{\left(\stackrel{n}{\sum _{i=1}}\mathbb{X}\left({\mu }_i,.\right)\left({\nu }_i-{\nu }_{i-1}\right)-\mathbb{Y}\left(.\right)\right)}^2\right]=0,$$

which can also be written as

$$\mathbb{Y}\left(.\right)=\underset{\mathrm{a}}{\stackrel{\mathrm{b}}{\int }}\mathbb{X}\left(s,.\right)\mathrm{d}s,(a.e.).$$

The notion of stochastic Riemann–Liouville fractional integrals were introduced by Hafiz in [38] as follows.

Definition 10

([38]). Let $\mathbb{X}:\mathcal{D}\times \mathrm{\Lambda }\to \mathbb{R}$ be a stochastic process such that $\mathbb{X}\in \mathbf{R}\left([\varphi ,\psi ]\right)$, where $\mathbf{R}\left([\varphi ,\psi ]\right)$ denotes the class of all mean-square Riemann integrable stochastic processes on the interval $[\varphi ,\psi ]$. The concept of stochastic mean-square Riemann–Liouville fractional integrals ${\mathcal{R}}_{{\varphi }^{+}}^{\alpha }$ and ${\mathcal{R}}_{{\psi }^{-}}^{\alpha }$ of $\mathbb{X}$ of order $\alpha >0$ is defined by

$${\mathcal{R}}_{{\varphi }^{+}}^{\alpha }\mathbb{X}\left(\varpi \right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}\stackrel{\varpi }{\underset{\varphi }{\int }}{\left(\varpi -u\right)}^{\alpha -1}\mathbb{X}\left(u,·\right)\mathrm{d}u(a.e.)\varpi >\varphi ,$$

and

$${\mathcal{R}}_{{\psi }^{-}}^{\alpha }\mathbb{X}\left(\varpi \right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}\stackrel{\psi }{\underset{\varpi }{\int }}{\left(u-\varpi \right)}^{\alpha -1}\mathbb{X}\left(u,·\right)\mathrm{d}u(a.e.)\varpi <\psi .$$

Definition 11

([18]). The integral representative of the hypergeometric function is defined as follows:

$${}_2\mathit{F}_1\left(\varphi ,\psi ,c;z\right)=\frac{1}{B\left(\psi ,c-\psi \right)}\underset{0}{\stackrel{1}{\int }}{\nu }^{\psi -1}{\left(1-\nu \right)}^{c-\psi -1}{\left(1-z\nu \right)}^{-a}\mathrm{d}\nu ,$$

where $c>\psi >0,\left|z\right|<1$ and $B\left(·,·\right)$ is the Beta function.

Definition 12

([39]). The integral representative of the Appell’s hypergeometric function ${\mathit{F}}_1$ is defined as follows:

$${\mathit{F}}_1\left(\varphi ,{\psi }_1,{\psi }_2,c,x,y\right)=\frac{\mathrm{\Gamma }\left(c\right)}{\mathrm{\Gamma }\left(\varphi \right)\mathrm{\Gamma }\left(c-\varphi \right)}\underset{0}{\stackrel{1}{\int }}{\nu }^{\varphi -1}{\left(1-\nu \right)}^{c-\varphi -1}{\left(1-x\nu \right)}^{-{\psi }_1}{\left(1-y\nu \right)}^{-{\psi }_2}\mathrm{d}\nu ,$$

with $c>\varphi >0,\left|x\right|<1$ and $\left|y\right|<1$.

Lemma 1

([40]). Assume that $\varphi ,\psi \ge 0$ and $p>0$. Then

$${\left(\varphi +\psi \right)}^p\le K_p\left({\varphi }^p+{\psi }^p\right),$$

where ${\mathcal{K}}_p=1$ if $0\le p\le 1$ and ${\mathcal{K}}_p=2^{p-1}$ if $p\ge 1$.

3. Auxiliary Results

This section is devoted to establishing an auxiliary lemma including technical estimates that serve as the backbone for the derivation of our main fractional inequalities.

Lemma 2.

For $\varphi ,\psi \ne 0$, $\alpha >0$ and $s\in (0,1]$, the following identities hold:

$$\begin{array}{cc}\hfill \underset{0}{\stackrel{\frac{1}{2}}{\int }}{\displaystyle \frac{{\nu }^{\alpha +s}}{{\left(\left(1-\nu \right)\varphi +\nu \psi \right)}^2}}\mathrm{d}\nu =& {\displaystyle \frac{{}_2\mathit{F}_1\left(2,1,\alpha +s+2;\frac{\psi -\varphi }{\varphi +\psi }\right)}{2^{\alpha +s-1}{\left(\varphi +\psi \right)}^2\left(\alpha +s+1\right)}}\hfill \\ \hfill =& {\mathcal{M}}_1\left(s,\alpha \right),\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill \underset{0}{\stackrel{\frac{1}{2}}{\int }}{\displaystyle \frac{{\nu }^{\alpha }{\left(1-\nu \right)}^s}{{\left(\nu \varphi +\left(1-\nu \right)\psi \right)}^2}}\mathrm{d}\nu =& {\displaystyle \frac{F_1\left(\alpha +1,-s,2,\alpha +2,\frac{1}{2},\frac{\psi -\varphi }{2\psi }\right)}{2^{\alpha +1}{\psi }^2\left(\alpha +1\right)}}\hfill \\ \hfill =& {\mathcal{M}}_2\left(s,\alpha \right),\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill \underset{\frac{1}{2}}{\stackrel{1}{\int }}{\displaystyle \frac{{\nu }^s-{\nu }^{\alpha +s}}{\left(\left(1-\nu \right)\varphi +t\psi \right)2}}\mathrm{d}\nu =& {\displaystyle \frac{1}{2{\psi }^2}}\left(F_1\left(1,-s,2,2,{\displaystyle \frac{1}{2}},{\displaystyle \frac{\psi -\varphi }{2\psi }}\right)-F_1\left(1,-\left(\alpha +s\right),2,2,{\displaystyle \frac{1}{2}},{\displaystyle \frac{\psi -\varphi }{2\psi }}\right)\right)\hfill \\ \hfill =& {\mathcal{M}}_3\left(s,\alpha \right),\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill \underset{\frac{1}{2}}{\stackrel{1}{\int }}{\displaystyle \frac{{\left(1-\nu \right)}^s-{\nu }^{\alpha }{\left(1-\nu \right)}^s}{{\left(\nu \varphi +\left(1-\nu \right)\psi \right)}^2}}\mathrm{d}\nu =& {\displaystyle \frac{1}{2{\psi }^2}}{F}_1\left(1,-s,2,2,{\displaystyle \frac{1}{2}},{\displaystyle \frac{\psi -\varphi }{2\psi }}\right)+{\displaystyle \frac{F_1\left(\alpha +1,-s,2,\alpha +2,\frac{1}{2},\frac{\psi -\varphi }{2\psi }\right)}{2^{\alpha +1}{\psi }^2\left(\alpha +1\right)}}\hfill \\ \hfill =& {\mathcal{M}}_4\left(s,\alpha \right),\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill \underset{0}{\stackrel{\frac{1}{2}}{\int }}{\displaystyle \frac{{\nu }^{\alpha }{\left(1-\nu \right)}^s}{{\left(\left(1-\nu \right)\varphi +\nu \psi \right)}^2}}\mathrm{d}\nu \le & {\displaystyle \frac{{}_2\mathit{F}_1\left(-2,1,\alpha +2;\frac{\psi -\varphi }{\varphi +\psi }\right)}{\left(\alpha +1\right){\left(\varphi +\psi \right)}^2{2}^{\alpha +s-1}}}+{\displaystyle \frac{B\left(s+1,\alpha +1\right){}_2\mathit{F}_1\left(-2,s+1,\alpha +s+2;\frac{\psi -\varphi }{\varphi +\psi }\right)}{{\left(\varphi +\psi \right)}^2{2}^{\alpha +s-1}}}\hfill \\ \hfill =& {\mathcal{M}}_5\left(s,\alpha \right),\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill \underset{0}{\stackrel{\frac{1}{2}}{\int }}{\displaystyle \frac{{\nu }^{\alpha +s}}{{\left(\nu \varphi +\left(1-\nu \right)\psi \right)}^2}}\mathrm{d}\nu =& {\displaystyle \frac{{}_2\mathit{F}_1\left(-2,\alpha +s+1,\alpha +s+2;\frac{\psi -\varphi }{2\psi }\right)}{\left(\alpha +s+1\right){\psi }^2{2}^{\alpha +s+1}}}\hfill \\ \hfill =& {\mathcal{M}}_6\left(s,\alpha \right),\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill \underset{\frac{1}{2}}{\stackrel{1}{\int }}{\displaystyle \frac{{\left(1-\nu \right)}^s-{\nu }^{\alpha }{\left(1-\nu \right)}^s}{{\left(\left(1-\nu \right)\varphi +\nu \psi \right)}^2}}\mathrm{d}\nu =& {\displaystyle \frac{1}{2^{s+1}{\psi }^2}}\left({\displaystyle \frac{{}_2\mathit{F}_1\left(2,s+1,s+2;\frac{\psi -\varphi }{2\psi }\right)}{s+1}}-{\displaystyle \frac{F_1\left(s+1,-\alpha ,2,s+2,\frac{1}{2},\frac{\psi -\varphi }{2\psi }\right)}{s+1}}\right)\hfill \\ \hfill =& {\mathcal{M}}_7\left(s,\alpha \right)\hfill \end{array}$$

(7)

and

$$\begin{array}{cc}\hfill \underset{\frac{1}{2}}{\stackrel{1}{\int }}{\displaystyle \frac{{\nu }^s-{\nu }^{\alpha +s}}{{\left(\nu \varphi +\left(1-\nu \right)\psi \right)}^2}}\mathrm{d}\nu \le & {\displaystyle \frac{1}{2^{s-1}{\left(\varphi +\psi \right)}^2}}\left({\displaystyle \frac{\left(\varphi +\psi \right)\left(4{\varphi }^2-{\left(\varphi +\psi \right)}^2\right)}{4{\varphi }^2\left(\psi -\varphi \right)}}+{\displaystyle \frac{{}_2\mathit{F}_1\left(2,s+1,s+2;\frac{\psi -\varphi }{\varphi +\psi }\right)}{s+1}}\right)\hfill \\ \hfill & -{\displaystyle \frac{{\mathcal{K}}_{\alpha +s}}{2^{\alpha +s-1}{\left(\varphi +\psi \right)}^2}}\left({\displaystyle \frac{\left(\varphi +\psi \right)\left(4{\varphi }^2-{\left(\varphi +\psi \right)}^2\right)}{4{\varphi }^2\left(\psi -\varphi \right)}}+{\displaystyle \frac{{}_2\mathit{F}_1\left(2,\alpha +s+1,\alpha +s+2;\frac{\psi -\varphi }{\varphi +\psi }\right)}{\alpha +s+1}}\right)\hfill \\ \hfill =& {\mathcal{M}}_8\left(s,\alpha \right).\hfill \end{array}$$

(8)

where ${\mathcal{K}}_{\alpha +s}$ is defined as in Lemma 1.

Here, we note that the hypergeometric function ${}_2\mathit{F}_1(.,.,.;.)$ and the Appell’s hypergeometric function $F_1(.,.,.;.)$ are both convergent since $\left|{\displaystyle \frac{\psi -\varphi }{\varphi +\psi }}\right|<1$ and $\left|{\displaystyle \frac{\psi -\varphi }{2\psi }}\right|<1$.

Proof. 

See Appendix A. □

4. Fractional Hermite–Hadamard Inequality for Harmonically $\mathbf{s}$-Convex Stochastic Processes

In this section, we begin by introducing the notion of harmonically s-convex stochastic processes in the second sense, then we present the associated stochastic Riemann–Liouville fractional Hermite–Hadamard inequality.

Definition 13.

A stochastic process $\mathbb{X}:\mathcal{D}\subset (0,\infty )\times \mathrm{\Lambda }\to \mathbb{R}$ is deemed to be harmonically s-convex stochastic process in the second sense for some fixed $s\in (0,1]$, if for all $u,v\in \mathcal{D}$ and every $\nu \in [0,1]$, one has

$$\mathbb{X}\left({\displaystyle \frac{uv}{(1-\nu )u+\nu v}},·\right)\le {\nu }^s\mathbb{X}(u,·)+{(1-\nu )}^s\mathbb{X}(v,·)(a.e.).$$

Theorem 4.

Let $0<\varphi <\psi$ and $\mathbb{X}:[\varphi ,\psi ]\times \mathrm{\Lambda }\to \mathbb{R}$ be a stochastic process that is mean-square integrable on $[\varphi ,\psi ]$. If $\mathbb{X}$ is exhibits harmonic s-convexity on $[\varphi ,\psi ]$, then the following Hermite–Hadamard inequality via stochastic Riemann–Liouville fractional integrals holds:

$$\begin{array}{cc}\hfill 2^{s-1}\mathbb{X}\left({\displaystyle \frac{2\varphi \psi }{\varphi +\psi }},·\right)\le & {\displaystyle \frac{\mathrm{\Gamma }\left(\alpha +1\right)}{2}}{\left({\displaystyle \frac{\varphi \psi }{\psi -\varphi }}\right)}^{\alpha }\left({\mathbf{J}}_{{\frac{1}{\varphi }}^{-}}^{\alpha }\left(\mathbb{X}\circ \mathrm{h}\right)\left({\displaystyle \frac{1}{\psi }},·\right)+{\mathbf{J}}_{{\frac{1}{\psi }}^{+}}^{\alpha }\left(\mathbb{X}\circ \mathrm{h}\right)\left({\displaystyle \frac{1}{\varphi }},·\right)\right)\hfill \\ \hfill \le & \left({\displaystyle \frac{\alpha }{\alpha +s}}+\alpha B(\alpha ,s+1)\right){\displaystyle \frac{\mathbb{X}(\varphi ,·)+\mathbb{X}(\psi ,·)}{2}},\hfill \end{array}$$

(9)

where $B(·,·)$ denotes the Beta function and $\mathrm{h}\left(u\right)={\displaystyle \frac{1}{u}}$.

Proof. 

The harmonic s-convexity of $\mathbb{X}$ allows us to write

$$\mathbb{X}\left({\displaystyle \frac{2uv}{u+v}}\right)\le {\displaystyle \frac{\mathbb{X}(u,·)+\mathbb{X}(v,·)}{2^s}}.$$

By making changes in variables $u={\displaystyle \frac{\varphi \psi }{\nu \varphi +(1-\nu )\psi }}$ and $v={\displaystyle \frac{\varphi \psi }{(1-\nu )\varphi +\nu \psi }}$, we obtain

$$2^s\mathbb{X}\left({\displaystyle \frac{2\varphi \psi }{\varphi +\psi }},·\right)\le \mathbb{X}\left({\displaystyle \frac{\varphi \psi }{\nu \varphi +(1-\nu )\psi }},·\right)+\mathbb{X}\left({\displaystyle \frac{\varphi \psi }{(1-\nu )\varphi +\nu \psi }},·\right).$$

(10)

Integrating the product of (10) and ${\nu }^{\alpha -1}$ over $[0,1]$ gives

$$\begin{array}{cc}\hfill 2^s\mathbb{X}\left({\displaystyle \frac{2\varphi \psi }{\varphi +\psi }},·\right)\le & \alpha \left[\underset{0}{\stackrel{1}{\int }}{\nu }^{\alpha -1}\mathbb{X}\left({\displaystyle \frac{\varphi \psi }{\nu \varphi +(1-\nu )\psi }},·\right)+\mathrm{d}\nu +\underset{0}{\stackrel{1}{\int }}{\nu }^{\alpha -1}\mathbb{X}\left({\displaystyle \frac{\varphi \psi }{(1-\nu )\varphi +\nu \psi }},·\right)\mathrm{d}\nu \right]\hfill \\ \hfill =& \alpha {\left({\displaystyle \frac{\varphi \psi }{\psi -\varphi }}\right)}^{\alpha }\left[\underset{\frac{1}{\psi }}{\stackrel{\frac{1}{\varphi }}{\int }}{\left(\mu -{\displaystyle \frac{1}{\psi }}\right)}^{\alpha -1}\mathbb{X}\left({\displaystyle \frac{1}{\mu }},·\right)\mathrm{d}\mu +\underset{\frac{1}{\psi }}{\stackrel{\frac{1}{\varphi }}{\int }}{\left({\displaystyle \frac{1}{\varphi }}-\mu \right)}^{\alpha -1}\mathbb{X}\left({\displaystyle \frac{1}{\mu }},·\right)\mathrm{d}\mu \right]\hfill \\ \hfill =& \mathrm{\Gamma }(\alpha +1){\left({\displaystyle \frac{\varphi \psi }{\psi -\varphi }}\right)}^{\alpha }\left({\mathbf{J}}_{{\frac{1}{\varphi }}^{-}}^{\alpha }\left(\mathbb{X}\circ \mathrm{h}\right)\left({\displaystyle \frac{1}{\psi }},·\right)+{\mathbf{J}}_{{\frac{1}{\psi }}^{+}}^{\alpha }\left(\mathbb{X}\circ \mathrm{h}\right)\left({\displaystyle \frac{1}{\varphi }},·\right)\right),\hfill \end{array}$$

which proves the first inequality in (9).

We proceed analogously with the second inequality. Starting from the harmonic s-convexity of $\mathbb{X}$, we obtain

$$\mathbb{X}\left({\displaystyle \frac{\varphi \psi }{\nu \varphi +(1-\nu )\psi }},·\right)+\mathbb{X}\left(\frac{\varphi \psi }{(1-\nu )\varphi +\nu \psi },·\right)\le \left({\nu }^s+{(1-\nu )}^s\right)\left[\mathbb{X}(\varphi ,·)+\mathbb{X}(\psi ,·)\right].$$

(11)

Integrating the product of (11) and ${\nu }^{\alpha -1}$ over $[0,1]$ gives

$$\begin{array}{cc}\hfill \mathrm{\Gamma }(\alpha +1){\left({\displaystyle \frac{\varphi \psi }{\psi -\varphi }}\right)}^{\alpha }\left({\mathbf{J}}_{{\frac{1}{\varphi }}^{-}}^{\alpha }\left(\mathbb{X}\circ \mathrm{h}\right)\left({\displaystyle \frac{1}{\psi }},·\right)+{\mathbf{J}}_{{\frac{1}{\psi }}^{+}}^{\alpha }\left(\mathbb{X}\circ \mathrm{h}\right)\left({\displaystyle \frac{1}{\varphi }},·\right)\right)\le & \alpha \left[\mathbb{X}(\varphi ,·)+\mathbb{X}(\psi ,·)\right]\underset{0}{\stackrel{1}{\int }}{\nu }^{\alpha -1}\left({\nu }^s+{(1-\nu )}^s\right)\mathrm{d}\nu \hfill \\ \hfill =& \left({\displaystyle \frac{\alpha }{\alpha +s}}+\alpha B(\alpha ,s+1)\right)\left[\mathbb{X}(\varphi ,·)+\mathbb{X}(\psi ,·)\right],\hfill \end{array}$$

which proves the second inequality in (9). □

Corollary 1.

If we attempt to take $s=1$, Theorem 4 yields the following fractional Hermite–Hadamard inequality for harmonically convex stochastic processes

$$\mathbb{X}\left({\displaystyle \frac{2\varphi \psi }{\varphi +\psi }},·\right)\le {\displaystyle \frac{\mathrm{\Gamma }\left(\alpha +1\right)}{2}}{\left({\displaystyle \frac{\varphi \psi }{\psi -\varphi }}\right)}^{\alpha }\left({\mathbf{J}}_{{\frac{1}{\varphi }}^{-}}^{\alpha }\left(\mathbb{X}\circ \mathrm{h}\right)\left({\displaystyle \frac{1}{\psi }},·\right)+{\mathbf{J}}_{{\frac{1}{\psi }}^{+}}^{\alpha }\left(\mathbb{X}\circ \mathrm{h}\right)\left({\displaystyle \frac{1}{\varphi }},·\right)\right)\le {\displaystyle \frac{\mathbb{X}(\varphi ,·)+\mathbb{X}(\psi ,·)}{2}}.$$

Remark 1.

By setting $\alpha =s=1$, Theorem 4 will be reduced to the result obtained by Okur et al. in ([22] Theorem 3.1).

5. Midpoint-Type Inequalities

We begin this section by introducing a new integral identity involving stochastic Riemann–Liouville fractional integrals, which we subsequently employ to derive a series of sharp midpoint-type inequalities in the fractional setting for mean-square differentiable, harmonically s-convex stochastic processes.

Lemma 3.

Let $\mathbb{X}:\mathcal{D}\times \mathrm{\Lambda }\to \mathbb{R}$ be a mean-square differentiable stochastic process on ${\mathcal{D}}^{\circ }$. If ${\mathbb{X}}^{\prime }\in \mathbf{R}\left([\varphi ,\psi ]\right)$ where $\varphi ,\psi \in {\mathcal{D}}^{\circ }$ with $0<\varphi <\psi$, then the following equality

$$\begin{array}{cc}\hfill & \mathbb{X}\left({\displaystyle \frac{2\varphi \psi }{\varphi +\psi }},·\right)-{\displaystyle \frac{\mathrm{\Gamma }\left(\alpha +1\right)}{2}}{\left({\displaystyle \frac{\varphi \psi }{\psi -\varphi }}\right)}^{\alpha }\left({\mathbf{J}}_{{\frac{1}{\varphi }}^{-}}^{\alpha }\left(\mathbb{X}\circ \mathrm{h}\right)\left({\displaystyle \frac{1}{\psi }},·\right)+{\mathbf{J}}_{{\frac{1}{\psi }}^{+}}^{\alpha }\left(\mathbb{X}\circ \mathrm{h}\right)\left({\displaystyle \frac{1}{\varphi }},·\right)\right)\hfill \\ \hfill =& {\displaystyle \frac{\varphi \psi \left(\psi -\varphi \right)}{2}}\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}{\nu }^{\alpha }\left({\displaystyle \frac{1}{{\left(\nu \varphi +\left(1-\nu \right)\psi \right)}^2}}{\mathbb{X}}^{\prime }\left({\displaystyle \frac{\varphi \psi }{\nu \varphi +\left(1-\nu \right)\psi }},·\right)-{\displaystyle \frac{1}{{\left(\left(1-\nu \right)\varphi +\nu \psi \right)}^2}}{\mathbb{X}}^{\prime }\left({\displaystyle \frac{\varphi \psi }{\left(1-\nu \right)\varphi +\nu \psi }},·\right)\right)\mathrm{d}\nu \right.\hfill \\ \hfill & +\left.\underset{\frac{1}{2}}{\stackrel{1}{\int }}\left(1-{\nu }^{\alpha }\right)\left({\displaystyle \frac{1}{{\left(\left(1-\nu \right)\varphi +\nu \psi \right)}^2}}{\mathbb{X}}^{\prime }\left({\displaystyle \frac{\varphi \psi }{\left(1-\nu \right)\varphi +\nu \psi }},·\right)-{\displaystyle \frac{1}{{\left(\nu \varphi +\left(1-\nu \right)\psi \right)}^2}}{\mathbb{X}}^{\prime }\left({\displaystyle \frac{\varphi \psi }{\nu \varphi +\left(1-\nu \right)\psi }},·\right)\right)\mathrm{d}\nu \right)\hfill \end{array}$$

holds, where $\mathrm{h}\left(u\right)={\displaystyle \frac{1}{u}}$.

Proof. 

Let

$$H_1=\underset{0}{\stackrel{\frac{1}{2}}{\int }}{\nu }^{\alpha }\left({\displaystyle \frac{1}{{\left(\nu \varphi +\left(1-\nu \right)\psi \right)}^2}}{\mathbb{X}}^{\prime }\left({\displaystyle \frac{\varphi \psi }{\nu \varphi +\left(1-\nu \right)\psi }},·\right)-{\displaystyle \frac{1}{{\left(\left(1-\nu \right)\varphi +\nu \psi \right)}^2}}{\mathbb{X}}^{\prime }\left({\displaystyle \frac{\varphi \psi }{\left(1-\nu \right)\varphi +\nu \psi }},·\right)\right)\mathrm{d}\nu$$

and

$$H_2=\underset{\frac{1}{2}}{\stackrel{1}{\int }}\left(1-{\nu }^{\alpha }\right)\left({\displaystyle \frac{1}{{\left(\left(1-\nu \right)\varphi +\nu \psi \right)}^2}}{\mathbb{X}}^{\prime }\left({\displaystyle \frac{\varphi \psi }{\left(1-\nu \right)\varphi +\nu \psi }},·\right)-{\displaystyle \frac{1}{{\left(\nu \varphi +\left(1-\nu \right)\psi \right)}^2}}{\mathbb{X}}^{\prime }\left({\displaystyle \frac{\varphi \psi }{\nu \varphi +\left(1-\nu \right)\psi }},·\right)\right)\mathrm{d}\nu .$$

Integrating by parts, $H_1$ we obtain

$$\begin{array}{cc}\hfill H_1=& \underset{0}{\stackrel{\frac{1}{2}}{\int }}{\nu }^{\alpha }\left({\displaystyle \frac{1}{{\left(\nu \varphi +\left(1-\nu \right)\psi \right)}^2}}{\mathbb{X}}^{\prime }\left({\displaystyle \frac{\varphi \psi }{\nu \varphi +\left(1-\nu \right)\psi }},·\right)-{\displaystyle \frac{1}{{\left(\left(1-\nu \right)\varphi +\nu \psi \right)}^2}}{\mathbb{X}}^{\prime }\left({\displaystyle \frac{\varphi \psi }{\left(1-\nu \right)\varphi +\nu \psi }},·\right)\right)\mathrm{d}\nu \hfill \\ \hfill =& {\left.{\displaystyle \frac{{\nu }^{\alpha }}{\varphi \psi \left(\psi -\varphi \right)}}\left(\mathbb{X}\left({\displaystyle \frac{\varphi \psi }{\left(1-\nu \right)\varphi +\nu \psi }},·\right)+\mathbb{X}\left({\displaystyle \frac{\varphi \psi }{\nu \varphi +\left(1-\nu \right)\psi }},·\right)\right)\right|}_0^{\frac{1}{2}}\hfill \\ \hfill & -{\displaystyle \frac{\alpha }{\varphi \psi \left(\psi -\varphi \right)}}\underset{0}{\stackrel{\frac{1}{2}}{\int }}{\nu }^{\alpha -1}\left(\mathbb{X}\left({\displaystyle \frac{\varphi \psi }{\left(1-\nu \right)\varphi +\nu \psi }},·\right)+\mathbb{X}\left({\displaystyle \frac{\varphi \psi }{\nu \varphi +\left(1-\nu \right)\psi }},·\right)\right)\mathrm{d}\nu \hfill \\ \hfill =& {\displaystyle \frac{1}{2^{\alpha -1}\varphi \psi \left(\psi -\varphi \right)}}\mathbb{X}\left({\displaystyle \frac{2\varphi \psi }{\varphi +\psi }},·\right)\hfill \\ \hfill & -{\displaystyle \frac{\alpha }{\varphi \psi \left(\psi -\varphi \right)}}\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}{\nu }^{\alpha -1}\mathbb{X}\left({\displaystyle \frac{\varphi \psi }{\left(1-\nu \right)\varphi +\nu \psi }},·\right)\mathrm{d}\nu +\underset{0}{\stackrel{\frac{1}{2}}{\int }}{\nu }^{\alpha -1}\mathbb{X}\left({\displaystyle \frac{\varphi \psi }{\nu \varphi +\left(1-\nu \right)\psi }},·\right)\mathrm{d}\nu \right).\hfill \end{array}$$

(12)

Similarly, we have

$$\begin{array}{cc}\hfill H_2=& \underset{\frac{1}{2}}{\stackrel{1}{\int }}\left(1-{\nu }^{\alpha }\right)\left({\displaystyle \frac{1}{{\left(\left(1-\nu \right)\varphi +\nu \psi \right)}^2}}{\mathbb{X}}^{\prime }\left({\displaystyle \frac{\varphi \psi }{\left(1-\nu \right)\varphi +\nu \psi }},·\right)-{\displaystyle \frac{1}{{\left(\nu \varphi +\left(1-\nu \right)\psi \right)}^2}}{\mathbb{X}}^{\prime }\left({\displaystyle \frac{\varphi \psi }{\nu \varphi +\left(1-\nu \right)\psi }},·\right)\right)\mathrm{d}\nu \hfill \\ \hfill =& {\left.-{\displaystyle \frac{1-{\nu }^{\alpha }}{\varphi \psi \left(\psi -\varphi \right)}}\left(\mathbb{X}\left({\displaystyle \frac{\varphi \psi }{\left(1-\nu \right)\varphi +\nu \psi }},·\right)+\mathbb{X}\left({\displaystyle \frac{\varphi \psi }{\nu \varphi +\left(1-\nu \right)\psi }},·\right)\right)\right|}_{{\displaystyle \frac{1}{2}}}^1\hfill \\ \hfill & -{\displaystyle \frac{\alpha }{\varphi \psi \left(\psi -\varphi \right)}}\underset{\frac{1}{2}}{\stackrel{1}{\int }}{\nu }^{\alpha -1}\left(\mathbb{X}\left({\displaystyle \frac{\varphi \psi }{\left(1-\nu \right)\varphi +\nu \psi }},·\right)+\mathbb{X}\left({\displaystyle \frac{\varphi \psi }{\nu \varphi +\left(1-\nu \right)\psi }},·\right)\right)\mathrm{d}\nu \hfill \\ \hfill =& {\displaystyle \frac{\varphi \psi }{\psi -\varphi }}\left(2-{\left({\displaystyle \frac{1}{2}}\right)}^{\alpha -1}\right)\mathbb{X}\left({\displaystyle \frac{2\varphi \psi }{\varphi +\psi }},·\right)\hfill \\ \hfill & -{\displaystyle \frac{\alpha }{\varphi \psi \left(\psi -\varphi \right)}}\left(\underset{\frac{1}{2}}{\stackrel{1}{\int }}{\nu }^{\alpha -1}\mathbb{X}\left({\displaystyle \frac{\varphi \psi }{\left(1-\nu \right)\varphi +\nu \psi }},·\right)\mathrm{d}\nu +\underset{\frac{1}{2}}{\stackrel{1}{\int }}{\nu }^{\alpha -1}\mathbb{X}\left({\displaystyle \frac{\varphi \psi }{\nu \varphi +\left(1-\nu \right)\psi }},·\right)\mathrm{d}\nu \right).\hfill \end{array}$$

(13)

Adding (12) and (13), we obtain

$$\begin{array}{cc}\hfill H_1+H_2=& {\displaystyle \frac{2}{\varphi \psi \left(\psi -\varphi \right)}}\mathbb{X}\left({\displaystyle \frac{2\varphi \psi }{\varphi +\psi }},·\right)\hfill \\ \hfill & -{\displaystyle \frac{\alpha }{\varphi \psi \left(\psi -\varphi \right)}}\left(\underset{0}{\stackrel{1}{\int }}{\nu }^{\alpha -1}\mathbb{X}\left({\displaystyle \frac{\varphi \psi }{\left(1-\nu \right)\varphi +\nu \psi }},·\right)\mathrm{d}\nu +\underset{0}{\stackrel{1}{\int }}{\nu }^{\alpha -1}\mathbb{X}\left({\displaystyle \frac{\varphi \psi }{\nu \varphi +\left(1-\nu \right)\psi }},·\right)\mathrm{d}\nu \right).\hfill \end{array}$$

(14)

Using the following change in variable $z={\displaystyle \frac{\varphi \psi }{\left(1-\nu \right)\varphi +\nu \psi }}$, we obtain

$$\begin{array}{cc}\hfill \underset{0}{\stackrel{1}{\int }}{\nu }^{\alpha -1}\mathbb{X}\left({\displaystyle \frac{\varphi \psi }{\left(1-\nu \right)\varphi +\nu \psi }},·\right)\mathrm{d}\nu =& {\left({\displaystyle \frac{\varphi \psi }{\psi -\varphi }}\right)}^{\alpha }\underset{\varphi }{\stackrel{\psi }{\int }}{\left({\displaystyle \frac{1}{z}}-{\displaystyle \frac{1}{\psi }}\right)}^{\alpha -1}\mathbb{X}\left(z,·\right){\displaystyle \frac{\mathrm{d}z}{z^2}}.\hfill \end{array}$$

(15)

Putting $u={\displaystyle \frac{1}{z}}$, (15) yields

$$\begin{array}{cc}\hfill \underset{0}{\stackrel{1}{\int }}{\nu }^{\alpha -1}\mathbb{X}\left({\displaystyle \frac{\varphi \psi }{\left(1-\nu \right)\varphi +\nu \psi }},·\right)\mathrm{d}\nu =& {\left({\displaystyle \frac{\varphi \psi }{\psi -\varphi }}\right)}^{\alpha }\underset{\frac{1}{\psi }}{\stackrel{\frac{1}{\varphi }}{\int }}{\left(u-{\displaystyle \frac{1}{\psi }}\right)}^{\alpha -1}\mathbb{X}\left({\displaystyle \frac{1}{u}},·\right)\mathrm{d}u.\hfill \end{array}$$

(16)

Similarly, we obtain

$$\begin{array}{cc}\hfill \underset{0}{\stackrel{1}{\int }}{\nu }^{\alpha -1}\mathbb{X}\left({\displaystyle \frac{\varphi \psi }{\nu \varphi +\left(1-\nu \right)\psi }},·\right)\mathrm{d}\nu =& {\left({\displaystyle \frac{\varphi \psi }{\psi -\varphi }}\right)}^{\alpha }\underset{\varphi }{\stackrel{\psi }{\int }}{\left({\displaystyle \frac{1}{\varphi }}-{\displaystyle \frac{1}{z}}\right)}^{\alpha -1}\mathbb{X}\left(v,·\right){\displaystyle \frac{dz}{z^2}}\hfill \\ \hfill =& {\left({\displaystyle \frac{\varphi \psi }{\psi -\varphi }}\right)}^{\alpha }\underset{\frac{1}{\psi }}{\stackrel{\frac{1}{\varphi }}{\int }}{\left({\displaystyle \frac{1}{\varphi }}-u\right)}^{\alpha -1}\mathbb{X}\left({\displaystyle \frac{1}{u}},·\right)du.\hfill \end{array}$$

(17)

Combining (14)–(17), we obtain

$$\begin{array}{cc}\hfill H_1+H_2=& {\displaystyle \frac{2}{\varphi \psi \left(\psi -\varphi \right)}}\mathbb{X}\left({\displaystyle \frac{2\varphi \psi }{\varphi +\psi }},·\right)\hfill \\ \hfill & -{\displaystyle \frac{\mathrm{\Gamma }\left(\alpha +1\right)}{\varphi \psi \left(\psi -\varphi \right)}}{\left({\displaystyle \frac{\varphi \psi }{\psi -\varphi }}\right)}^{\alpha }\left(\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}\underset{\frac{1}{\psi }}{\stackrel{\frac{1}{\varphi }}{\int }}{\left(u-{\displaystyle \frac{1}{\psi }}\right)}^{\alpha -1}\mathbb{X}\left({\displaystyle \frac{1}{u}},·\right)du+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}\underset{\frac{1}{\psi }}{\stackrel{\frac{1}{\varphi }}{\int }}{\left({\displaystyle \frac{1}{\varphi }}-u\right)}^{\alpha -1}\mathbb{X}\left(\frac{1}{u}\right)du\right).\hfill \end{array}$$

(18)

Multiplying (18) by ${\displaystyle \frac{\varphi \psi \left(\psi -\varphi \right)}{2}}$, we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{\varphi \psi \left(\psi -\varphi \right)}{2}}\left(H_1+H_2\right)=& \mathbb{X}\left({\displaystyle \frac{2\varphi \psi }{\varphi +\psi }},·\right)\hfill \\ \hfill & -{\displaystyle \frac{\mathrm{\Gamma }\left(\alpha +1\right)}{2}}{\left(\frac{\varphi \psi }{\psi -\varphi }\right)}^{\alpha }\left({\mathbf{J}}_{{\frac{1}{\varphi }}^{-}}^{\alpha }\left(\mathbb{X}\circ \mathrm{h}\right)\left({\displaystyle \frac{1}{\psi }},·\right)+{\mathbf{J}}_{{\frac{1}{\psi }}^{+}}^{\alpha }\left(\mathbb{X}\circ \mathrm{h}\right)\left({\displaystyle \frac{1}{\varphi }},·\right)\right),\hfill \end{array}$$

which is the desired result. □

Theorem 5.

Let $0<\varphi <\psi$ and $\mathbb{X}:[\varphi ,\psi ]\times \mathrm{\Lambda }\to \mathbb{R}$ be a mean-square differentiable stochastic process on ${\mathcal{D}}^{\circ }$. If $\left|{\mathbb{X}}^{\prime }\right|$ is a harmonically s-convex stochastic process on $\left[\varphi ,\psi \right]$, then we have

$$\begin{array}{cc}\hfill & \left|\mathbb{X}\left({\displaystyle \frac{2\varphi \psi }{\varphi +\psi }},·\right)-{\displaystyle \frac{\mathrm{\Gamma }\left(\alpha +1\right)}{2}}{\left({\displaystyle \frac{\varphi \psi }{\psi -\varphi }}\right)}^{\alpha }\left({\mathbf{J}}_{{\frac{1}{\varphi }}^{-}}^{\alpha }\left(\mathbb{X}\circ \mathrm{h}\right)\left({\displaystyle \frac{1}{\psi }},·\right)+{\mathbf{J}}_{{\frac{1}{\psi }}^{+}}^{\alpha }\left(\mathbb{X}\circ \mathrm{h}\right)\left({\displaystyle \frac{1}{\varphi }},·\right)\right)\right|\hfill \\ \hfill \le & {\displaystyle \frac{\varphi \psi \left(\psi -\varphi \right)}{2}}\left[\left({\mathcal{M}}_1(s,\alpha )+{\mathcal{M}}_2(s,\alpha )+{\mathcal{M}}_3(s,\alpha )+{\mathcal{M}}_4(s,\alpha )\right)\left|{\mathbb{X}}^{\prime }\left(\varphi ,·\right)\right|\right.\hfill \\ \hfill & \left.+\left({\mathcal{M}}_5(s,\alpha )+{\mathcal{M}}_6(s,\alpha )+{\mathcal{M}}_7(s,\alpha )+{\mathcal{M}}_8(s,\alpha )\right)\left|{\mathbb{X}}^{\prime }\left(\psi ,·\right)\right|\right],\hfill \end{array}$$

where ${\mathcal{M}}_k(s,\alpha )$ ($k=1,\dots ,8$) are defined as in (1)–(8), respectively.

Proof. 

From Lemma 3, absolute value and harmonically s-convexity of $\left|{\mathbb{X}}^{\prime }\right|$, we have

$$\begin{array}{cc}\hfill & \left|\mathbb{X}\left({\displaystyle \frac{2\varphi \psi }{\varphi +\psi }},·\right)-{\displaystyle \frac{\mathrm{\Gamma }\left(\alpha +1\right)}{2}}{\left({\displaystyle \frac{\varphi \psi }{\psi -\varphi }}\right)}^{\alpha }\left({\mathbf{J}}_{{\frac{1}{\varphi }}^{-}}^{\alpha }\left(\mathbb{X}\circ \mathrm{h}\right)\left({\displaystyle \frac{1}{\psi }},·\right)+{\mathbf{J}}_{{\frac{1}{\psi }}^{+}}^{\alpha }\left(\mathbb{X}\circ \mathrm{h}\right)\left({\displaystyle \frac{1}{\varphi }},·\right)\right)\right|\hfill \\ \hfill \le & {\displaystyle \frac{\varphi \psi \left(\psi -\varphi \right)}{2}}\left[\underset{0}{\stackrel{\frac{1}{2}}{\int }}{\nu }^{\alpha }\left({\displaystyle \frac{1}{{\left(\left(1-\nu \right)\varphi +\nu \psi \right)}^2}}{\mathbb{X}}^{\prime }\left({\displaystyle \frac{\varphi \psi }{\left(1-\nu \right)\varphi +\nu \psi }},·\right)+{\displaystyle \frac{1}{{\left(\nu \varphi +\left(1-\nu \right)\psi \right)}^2}}{\mathbb{X}}^{\prime }\left({\displaystyle \frac{\varphi \psi }{\nu \varphi +\left(1-\nu \right)\psi }},·\right)\right)\mathrm{d}\nu \right.\hfill \\ \hfill & +\left.\underset{\frac{1}{2}}{\stackrel{1}{\int }}\left(1-{\nu }^{\alpha }\right)\left({\displaystyle \frac{1}{{\left(\left(1-\nu \right)\varphi +\nu \psi \right)}^2}}{\mathbb{X}}^{\prime }\left({\displaystyle \frac{\varphi \psi }{\left(1-\nu \right)\varphi +\nu \psi }},·\right)+{\displaystyle \frac{1}{{\left(\nu \varphi +\left(1-\nu \right)\psi \right)}^2}}{\mathbb{X}}^{\prime }\left({\displaystyle \frac{\varphi \psi }{\nu \varphi +\left(1-\nu \right)\psi }},·\right)\right)\mathrm{d}\nu \right]\hfill \\ \hfill \le & {\displaystyle \frac{\varphi \psi \left(\psi -\varphi \right)}{2}}\hfill \\ \hfill & \times \left[\underset{0}{\stackrel{\frac{1}{2}}{\int }}\left({\displaystyle \frac{{\nu }^{\alpha }}{{\left(\left(1-\nu \right)\varphi +\nu \psi \right)}^2}}\left({\nu }^s\left|{\mathbb{X}}^{\prime }\left(\varphi ,·\right)\right|+{\left(1-\nu \right)}^s\left|{\mathbb{X}}^{\prime }\left(\psi \right)\right|\right)+{\displaystyle \frac{{\nu }^{\alpha }}{{\left(\nu \varphi +\left(1-\nu \right)\psi \right)}^2}}\left({\left(1-\nu \right)}^s\left|{\mathbb{X}}^{\prime }\left(\varphi \right)\right|+{\nu }^s\left|{\mathbb{X}}^{\prime }\left(\psi \right)\right|\right)\right)\mathrm{d}\nu \right.\hfill \\ \hfill & +\left.\underset{\frac{1}{2}}{\stackrel{1}{\int }}\left({\displaystyle \frac{1-{\nu }^{\alpha }}{{\left(\left(1-\nu \right)\varphi +\nu \psi \right)}^2}}\left({\nu }^s\left|{\mathbb{X}}^{\prime }\left(\varphi ,·\right)\right|+{\left(1-\nu \right)}^s\left|{\mathbb{X}}^{\prime }\left(\psi ,·\right)\right|\right)+{\displaystyle \frac{1-{\nu }^{\alpha }}{{\left(\nu \varphi +\left(1-\nu \right)\psi \right)}^2}}\left({\left(1-\psi \right)}^s\left|{\mathbb{X}}^{\prime }\left(\varphi ,·\right)\right|+{\nu }^s\left|{\mathbb{X}}^{\prime }\left(\psi ,·\right)\right|\right)\right)\mathrm{d}\nu \right]\hfill \\ \hfill \le & {\displaystyle \frac{\varphi \psi \left(\psi -\varphi \right)}{2}}\hfill \\ \hfill & \times \left[\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}{\displaystyle \frac{{\nu }^{\alpha +s}}{{\left(\left(1-\nu \right)\varphi +\nu \psi \right)}^2}}\mathrm{d}\nu +\underset{0}{\stackrel{\frac{1}{2}}{\int }}{\displaystyle \frac{{\nu }^{\alpha }{\left(1-\nu \right)}^s}{{\left(\nu \varphi +\left(1-\nu \right)\psi \right)}^2}}\mathrm{d}\nu +\underset{\frac{1}{2}}{\stackrel{1}{\int }}{\displaystyle \frac{{\nu }^s-{\nu }^{\alpha +s}}{{\left(\left(1-\nu \right)\varphi +\nu \psi \right)}^2}}\mathrm{d}\nu +\underset{\frac{1}{2}}{\stackrel{1}{\int }}{\displaystyle \frac{{\left(1-\nu \right)}^s-{\nu }^{\alpha }{\left(1-\nu \right)}^s}{{\left(\nu \varphi +\left(1-\nu \right)\psi \right)}^2}}\mathrm{d}\nu \right)\left|{\mathbb{X}}^{\prime }\left(\varphi ,·\right)\right|\right.\hfill \\ \hfill & +\left.\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}{\displaystyle \frac{{\nu }^{\alpha }{\left(1-\nu \right)}^s}{{\left(\left(1-\nu \right)\varphi +\nu \psi \right)}^2}}\mathrm{d}\nu +\underset{0}{\stackrel{\frac{1}{2}}{\int }}{\displaystyle \frac{{\nu }^{\alpha +s}}{{\left(\nu \varphi +\left(1-\nu \right)\psi \right)}^2}}\mathrm{d}\nu +\underset{\frac{1}{2}}{\stackrel{1}{\int }}{\displaystyle \frac{{\left(1-\nu \right)}^s-{\nu }^{\alpha }{\left(1-\nu \right)}^s}{{\left(\left(1-\nu \right)\varphi +\nu \psi \right)}^2}}\mathrm{d}\nu +\underset{\frac{1}{2}}{\stackrel{1}{\int }}{\displaystyle \frac{{\nu }^s-{\nu }^{\alpha +s}}{{\left(\nu \varphi +\left(1-\nu \right)\psi \right)}^2}}\mathrm{d}\nu \right)\left|{\mathbb{X}}^{\prime }\left(\psi ,·\right)\right|\right]\hfill \\ \hfill \le & {\displaystyle \frac{\varphi \psi \left(\psi -\varphi \right)}{2}}\left[\left({\mathcal{M}}_1(s,\alpha )+{\mathcal{M}}_2(s,\alpha )+{\mathcal{M}}_3(s,\alpha )+{\mathcal{M}}_4(s,\alpha )\right)\left|{\mathbb{X}}^{\prime }\left(\varphi ,·\right)\right|\right.\hfill \\ \hfill & \left.+\left({\mathcal{M}}_5(s,\alpha )+{\mathcal{M}}_6(s,\alpha )+{\mathcal{M}}_7(s,\alpha )+{\mathcal{M}}_8(s,\alpha )\right)\left|{\mathbb{X}}^{\prime }\left(\psi ,·\right)\right|\right],\hfill \end{array}$$

where we have used (1)–(8).

The proof is completed. □

Corollary 2.

In Theorem 5, if we take $s=1$, we obtain the following fractional midpoint-type inequality for mean-square differentiable harmonically convex stochastic processes:

$$\begin{array}{cc}\hfill & \left|\mathbb{X}\left({\displaystyle \frac{2\varphi \psi }{\varphi +\psi }},·\right)-{\displaystyle \frac{\mathrm{\Gamma }\left(\alpha +1\right)}{2}}{\left({\displaystyle \frac{\varphi \psi }{\psi -\varphi }}\right)}^{\alpha }\left({\mathbf{J}}_{{\frac{1}{\varphi }}^{-}}^{\alpha }\left(\mathbb{X}\circ \mathrm{h}\right)\left({\displaystyle \frac{1}{\psi }},·\right)+{\mathbf{J}}_{{\frac{1}{\psi }}^{+}}^{\alpha }\left(\mathbb{X}\circ \mathrm{h}\right)\left({\displaystyle \frac{1}{\varphi }},·\right)\right)\right|\hfill \\ \hfill \le & {\displaystyle \frac{\varphi \psi \left(\psi -\varphi \right)}{2}}\left[\left({\mathcal{M}}_1(1,\alpha )+{\mathcal{M}}_2(1,\alpha )+{\mathcal{M}}_3(1,\alpha )+{\mathcal{M}}_4(1,\alpha )\right)\left|{\mathbb{X}}^{\prime }\left(\varphi ,·\right)\right|\right.\hfill \\ \hfill & \left.+\left({\mathcal{M}}_5(1,\alpha )+{\mathcal{M}}_6(1,\alpha )+{\mathcal{M}}_7(1,\alpha )+{\mathcal{M}}_8(1,\alpha )\right)\left|{\mathbb{X}}^{\prime }\left(\psi ,·\right)\right|\right],\hfill \end{array}$$

where ${\mathcal{M}}_k(1,\alpha )$ ($k=1,\dots ,8$) are defined as in (1)–(8), respectively.

Corollary 3.

In Theorem 5, if we take $\alpha =1$, we obtain the following midpoint-type inequality for mean-square differentiable harmonically s-convex stochastic processes:

$$\begin{array}{cc}\hfill & \left|\mathbb{X}\left({\displaystyle \frac{2\varphi \psi }{\varphi +\psi }},·\right)-{\displaystyle \frac{\varphi \psi }{\psi -\varphi }}\underset{\varphi }{\stackrel{\psi }{\int }}{\displaystyle \frac{\mathbb{X}\left(u,·\right)}{u^2}}du\right|\hfill \\ \hfill \le & {\displaystyle \frac{\varphi \psi \left(\psi -\varphi \right)}{2}}\left[\left({\mathcal{M}}_1(s,1)+{\mathcal{M}}_2(s,1)+{\mathcal{M}}_3(s,1)+{\mathcal{M}}_4(s,1)\right)\left|{\mathbb{X}}^{\prime }\left(\varphi ,·\right)\right|\right.\hfill \\ \hfill & \left.+\left({\mathcal{M}}_5(s,1)+{\mathcal{M}}_6(s,1)+{\mathcal{M}}_7(s,1)+{\mathcal{M}}_8(s,1)\right)\left|{\mathbb{X}}^{\prime }\left(\psi ,·\right)\right|\right],\hfill \end{array}$$

where ${\mathcal{M}}_k(s,1)$ ($k=1,\dots ,8$) are defined as in (1)–(8), respectively.

Corollary 4.

In Theorem 5, if we take $\alpha =s=1$, we obtain the following midpoint-type inequality for mean-square differentiable harmonically convex stochastic processes:

$$\begin{array}{cc}\hfill & \left|\mathbb{X}\left({\displaystyle \frac{2\varphi \psi }{\varphi +\psi }},·\right)-{\displaystyle \frac{\varphi \psi }{\psi -\varphi }}\underset{\varphi }{\stackrel{\psi }{\int }}{\displaystyle \frac{\mathbb{X}\left(u,·\right)}{u^2}}du\right|\hfill \\ \hfill \le & {\displaystyle \frac{\varphi \psi \left(\psi -\varphi \right)}{2}}\left(\left({\displaystyle \frac{2(\varphi +\psi )}{{\left(\psi -\varphi \right)}^3}}ln{\displaystyle \frac{2\psi }{\varphi +\psi }}+{\displaystyle \frac{4\varphi }{{\left(\psi -\varphi \right)}^3}}ln{\displaystyle \frac{2\varphi }{\varphi +\psi }}\right)\left|{\mathbb{X}}^{\prime }\left(\varphi ,·\right)\right|\right.\hfill \\ \hfill & +\left.\left({\displaystyle \frac{2(\varphi +\psi )}{{\left(\psi -\varphi \right)}^3}}ln{\displaystyle \frac{\varphi +\psi }{2\psi }}+{\displaystyle \frac{4\psi }{{\left(\psi -\varphi \right)}^3}}ln{\displaystyle \frac{\varphi +\psi }{2\psi }}\right)\left|{\mathbb{X}}^{\prime }\left(\psi ·\right)\right|\right).\hfill \end{array}$$

Theorem 6.

Let $0<\varphi <\psi$ and $\mathbb{X}:[\varphi ,\psi ]\times \mathrm{\Lambda }\to \mathbb{R}$ be a mean-square differentiable stochastic process on ${\mathcal{D}}^{\circ }$. If ${\left|{\mathbb{X}}^{\prime }\right|}^q$ is a harmonically s-convex stochastic process on $\left[\varphi ,\psi \right]$ for $s\in (0,1]$ and $q>0$ with ${\displaystyle \frac{1}{q}}+{\displaystyle \frac{1}{p}}=1$, then for $\alpha >0$, we have

$$\begin{array}{cc}\hfill & \left|\mathbb{X}\left({\displaystyle \frac{2\varphi \psi }{\varphi +\psi }},·\right)-{\displaystyle \frac{\mathrm{\Gamma }\left(\alpha +1\right)}{2}}{\left({\displaystyle \frac{\varphi \psi }{\psi -\varphi }}\right)}^{\alpha }\left({\mathbf{J}}_{{\frac{1}{\varphi }}^{-}}^{\alpha }\left(\mathbb{X}\circ \mathrm{h}\right)\left({\displaystyle \frac{1}{\psi }},·\right)+{\mathbf{J}}_{{\frac{1}{\psi }}^{+}}^{\alpha }\left(\mathbb{X}\circ \mathrm{h}\right)\left({\displaystyle \frac{1}{\varphi }},·\right)\right)\right|\hfill \\ \hfill \le & {\displaystyle \frac{\varphi \psi \left(\psi -\varphi \right)}{2{\left(s+1\right)}^{\frac{1}{q}}}}\left(\left({\left({\displaystyle \frac{{}_2\mathit{F}_1\left(2p,1,\alpha p+2;\frac{\psi -\varphi }{\varphi +\psi }\right)}{2^{\alpha p-p+1}{\left(\varphi +\psi \right)}^{2p}\left(\alpha p+1\right)}}\right)}^{\frac{1}{p}}+{\left({\displaystyle \frac{1-2^{2p-\alpha -1}}{\left(\psi -\varphi \right){\varphi }^{2p-1}}}+{\displaystyle \frac{2^{2p-\alpha -1}-2^{2p-1}}{\left(\psi -\varphi \right){\left(\varphi +\psi \right)}^{2p-1}}}\right)}^{\frac{1}{p}}\right)\right.\hfill \\ \hfill & \times {\left({\displaystyle \frac{2^{s+1}-1}{2^{s+1}}}{\left|{\mathbb{X}}^{\prime }\left(\varphi ,·\right)\right|}^q+{\displaystyle \frac{1}{2^{s+1}}}{\left|{\mathbb{X}}^{\prime }\left(\psi ,·\right)\right|}^q\right)}^{\frac{1}{q}}\hfill \\ \hfill & +\left({\left({\displaystyle \frac{{}_2\mathit{F}_1\left(2p,\alpha p+1,\alpha p+2;\frac{\psi -\varphi }{2\psi }\right)}{2^{\alpha p+1}{\psi }^{2p}\left(\alpha p+1\right)}}\right)}^{\frac{1}{p}}+{\left({\displaystyle \frac{2^{2p-1}{\psi }^{2p-1}-{\left(\varphi +\psi \right)}^{2p-1}}{\left(2p-1\right)\left(\psi -\varphi \right){\psi }^{2p-1}{\left(\varphi +\psi \right)}^{2p-1}}}-{\displaystyle \frac{F_1\left(1,-\alpha ,2p,2,\frac{1}{2},\frac{\psi -\varphi }{2\psi }\right)}{2{\psi }^{2p}}}\right)}^{\frac{1}{p}}\right)\hfill \\ \hfill & \times \left.{\left({\displaystyle \frac{1}{2^{s+1}}}{\left|{\mathbb{X}}^{\prime }\left(\varphi ,·\right)\right|}^q+{\displaystyle \frac{2^{s+1}-1}{2^{s+1}}}{\left|{\mathbb{X}}^{\prime }\left(\psi ,·\right)\right|}^q\right)}^{\frac{1}{q}}\right),\hfill \end{array}$$

where ${}_2\mathit{F}_1$ and ${\mathit{F}}_1$ are the hypergeometric and the Appell’s hypergeometric functions.

Proof. 

From Lemma 3, absolute value, Hölder’s inequality, and harmonically s-convexity of ${\left|{\mathbb{X}}^{\prime }\right|}^q$, we have

$$\begin{array}{cc}\hfill & \left|\mathbb{X}\left({\displaystyle \frac{2\varphi \psi }{\varphi +\psi }},·\right)-{\displaystyle \frac{\mathrm{\Gamma }\left(\alpha +1\right)}{2}}{\left({\displaystyle \frac{\varphi \psi }{\psi -\varphi }}\right)}^{\alpha }\left({\mathbf{J}}_{{\frac{1}{\varphi }}^{-}}^{\alpha }\left(\mathbb{X}\circ \mathrm{h}\right)\left({\displaystyle \frac{1}{\psi }},·\right)+{\mathbf{J}}_{{\frac{1}{\psi }}^{+}}^{\alpha }\left(\mathbb{X}\circ \mathrm{h}\right)\left({\displaystyle \frac{1}{\varphi }},·\right)\right)\right|\hfill \\ \hfill \le & {\displaystyle \frac{\varphi \psi \left(\psi -\varphi \right)}{2}}\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}{\nu }^{\alpha }\left({\displaystyle \frac{1}{{\left(\left(1-\nu \right)\varphi +\nu \psi \right)}^2}}\left|{\mathbb{X}}^{\prime }\left({\displaystyle \frac{\varphi \psi }{\left(1-\nu \right)\varphi +\nu \psi }},·\right)\right|+{\displaystyle \frac{1}{{\left(\nu \varphi +\left(1-\nu \right)\psi \right)}^2}}\left|{\mathbb{X}}^{\prime }\left({\displaystyle \frac{\varphi \psi }{\nu \varphi +\left(1-\nu \right)\psi }},·\right)\right|\right)\mathrm{d}\nu \right.\hfill \\ \hfill & +\left.\underset{\frac{1}{2}}{\stackrel{1}{\int }}\left(1-{\nu }^{\alpha }\right)\left({\displaystyle \frac{1}{{\left(\left(1-\nu \right)\varphi +\nu \psi \right)}^2}}\left|{\mathbb{X}}^{\prime }\left({\displaystyle \frac{\varphi \psi }{\left(1-\nu \right)\varphi +\nu \psi }}\right)\right|+{\displaystyle \frac{1}{{\left(\nu \varphi +\left(1-\nu \right)\psi \right)}^2}}\left|{\mathbb{X}}^{\prime }\left({\displaystyle \frac{\varphi \psi }{\nu \varphi +\left(1-\nu \right)\psi }}\right)\right|\right)\mathrm{d}\nu \right)\hfill \\ \hfill \le & {\displaystyle \frac{\varphi \psi \left(\psi -\varphi \right)}{2}}\left({\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}{\displaystyle \frac{{\nu }^{\alpha p}}{{\left(\left(1-\nu \right)\varphi +\nu \psi \right)}^{2p}}}\mathrm{d}\nu \right)}^{\frac{1}{p}}{\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}{\left|{\mathbb{X}}^{\prime }\left({\displaystyle \frac{\varphi \psi }{\left(1-\nu \right)\varphi +\nu \psi }}\right)\right|}^q\mathrm{d}\nu \right)}^{\frac{1}{q}}\right.\hfill \\ \hfill & +{\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}{\displaystyle \frac{{\nu }^{\alpha p}}{{\left(\nu \varphi +\left(1-\nu \right)\psi \right)}^{2p}}}\mathrm{d}\nu \right)}^{\frac{1}{p}}{\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}{\left|{\mathbb{X}}^{\prime }\left({\displaystyle \frac{\varphi \psi }{\nu \varphi +\left(1-\nu \right)\psi }}\right)\right|}^q\mathrm{d}\nu \right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(\underset{\frac{1}{2}}{\stackrel{1}{\int }}{\displaystyle \frac{{\left(1-{\nu }^{\alpha }\right)}^p}{{\left(\left(1-\nu \right)\varphi +\nu \psi \right)}^{2p}}}\mathrm{d}\nu \right)}^{\frac{1}{p}}{\left(\underset{\frac{1}{2}}{\stackrel{1}{\int }}{\left|{\mathbb{X}}^{\prime }\left({\displaystyle \frac{\varphi \psi }{\left(1-\nu \right)\varphi +\nu \psi }}\right)\right|}^q\mathrm{d}\nu \right)}^{\frac{1}{q}}\hfill \\ \hfill & +\left.{\left(\underset{\frac{1}{2}}{\stackrel{1}{\int }}{\displaystyle \frac{{\left(1-{\nu }^{\alpha }\right)}^p}{{\left(\nu \varphi +\left(1-\nu \right)\psi \right)}^{2p}}}\mathrm{d}\nu \right)}^{\frac{1}{p}}{\left(\underset{\frac{1}{2}}{\stackrel{1}{\int }}{\left|{\mathbb{X}}^{\prime }\left({\displaystyle \frac{\varphi \psi }{\nu \varphi +\left(1-\nu \right)\psi }}\right)\right|}^q\mathrm{d}\nu \right)}^{\frac{1}{q}}\right)\hfill \\ \hfill \le & {\displaystyle \frac{\varphi \psi \left(\psi -\varphi \right)}{2}}\left({\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}{\displaystyle \frac{{\nu }^{\alpha p}}{{\left(\left(1-\nu \right)\varphi +\nu \psi \right)}^{2p}}}\mathrm{d}\nu \right)}^{\frac{1}{p}}{\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}\left({\left(1-\nu \right)}^s{\left|{\mathbb{X}}^{\prime }\left(\varphi ,·\right)\right|}^q+{\nu }^s{\left|{\mathbb{X}}^{\prime }\left(\psi \right)\right|}^q\right)\mathrm{d}\nu \right)}^{\frac{1}{q}}\right.\hfill \\ \hfill & +{\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}{\displaystyle \frac{{\nu }^{\alpha p}}{{\left(\nu \varphi +\left(1-\nu \right)\psi \right)}^{2p}}}\mathrm{d}\nu \right)}^{\frac{1}{p}}{\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}\left({\nu }^s{\left|{\mathbb{X}}^{\prime }\left(\varphi ,·\right)\right|}^q+{\left(1-\nu \right)}^s{\left|{\mathbb{X}}^{\prime }\left(\psi ,·\right)\right|}^q\right)\mathrm{d}\nu \right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(\underset{\frac{1}{2}}{\stackrel{1}{\int }}{\displaystyle \frac{{\left(1-{\nu }^{\alpha }\right)}^p}{{\left(\left(1-\nu \right)\varphi +\nu \psi \right)}^{2p}}}\mathrm{d}\nu \right)}^{\frac{1}{p}}{\left(\underset{\frac{1}{2}}{\stackrel{1}{\int }}\left({\left(1-\nu \right)}^s{\left|{\mathbb{X}}^{\prime }\left(\varphi ,·\right)\right|}^q+{\nu }^s{\left|{\mathbb{X}}^{\prime }\left(\psi ,·\right)\right|}^q\right)\mathrm{d}\nu \right)}^{\frac{1}{q}}\hfill \\ \hfill & +\left.{\left(\underset{\frac{1}{2}}{\stackrel{1}{\int }}{\displaystyle \frac{{\left(1-{\nu }^{\alpha }\right)}^p}{{\left(\nu \varphi +\left(1-\nu \right)\psi \right)}^{2p}}}\mathrm{d}\nu \right)}^{\frac{1}{p}}{\left(\underset{\frac{1}{2}}{\stackrel{1}{\int }}\left({\nu }^s{\left|{\mathbb{X}}^{\prime }\left(\varphi ,·\right)\right|}^q+{\left(1-\nu \right)}^s{\left|{\mathbb{X}}^{\prime }\left(\psi ,·\right)\right|}^q\right)\mathrm{d}\nu \right)}^{\frac{1}{q}}\right)\hfill \\ \hfill \le & {\displaystyle \frac{\varphi \psi \left(\psi -\varphi \right)}{2{\left(s+1\right)}^{\frac{1}{q}}}}\left({\left({\displaystyle \frac{{}_2\mathit{F}_1\left(2p,1,\alpha p+2;\frac{\psi -\varphi }{\varphi +\psi }\right)}{2^{\alpha p-p+1}{\left(\varphi +\psi \right)}^{2p}\left(\alpha p+1\right)}}\right)}^{\frac{1}{p}}{\left({\displaystyle \frac{2^{s+1}-1}{2^{s+1}}}{\left|{\mathbb{X}}^{\prime }\left(\varphi ,·\right)\right|}^q+{\displaystyle \frac{1}{2^{s+1}}}{\left|{\mathbb{X}}^{\prime }\left(\psi ,·\right)\right|}^q\right)}^{\frac{1}{q}}\right.\hfill \\ \hfill & +{\left({\displaystyle \frac{{}_2\mathit{F}_1\left(2p,\alpha p+1,\alpha p+2;\frac{\psi -\varphi }{2\psi }\right)}{2^{\alpha p+1}{\psi }^{2p}\left(\alpha p+1\right)}}\right)}^{\frac{1}{p}}{\left({\displaystyle \frac{1}{2^{s+1}}}{\left|{\mathbb{X}}^{\prime }\left(\varphi ,·\right)\right|}^q+{\displaystyle \frac{2^{s+1}-1}{2^{s+1}}}{\left|{\mathbb{X}}^{\prime }\left(\psi ,·\right)\right|}^q\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left({\displaystyle \frac{2^{2p-1}{\psi }^{2p-1}-{\left(\varphi +\psi \right)}^{2p-1}}{\left(2p-1\right)\left(\psi -\varphi \right){\psi }^{2p-1}{\left(\varphi +\psi \right)}^{2p-1}}}-{\displaystyle \frac{F_1\left(1,-\alpha ,2p,2,\frac{1}{2},\frac{\psi -\varphi }{2\psi }\right)}{2{\psi }^{2p}}}\right)}^{\frac{1}{p}}{\left({\displaystyle \frac{1}{2^{s+1}}}{\left|{\mathbb{X}}^{\prime }\left(\varphi ,\right)\right|}^q+{\displaystyle \frac{2^{s+1}-1}{2^{s+1}}}{\left|{\mathbb{X}}^{\prime }\left(\psi ,·\right)\right|}^q\right)}^{\frac{1}{q}}\hfill \\ \hfill & +\left.{\left({\displaystyle \frac{1}{\psi -\varphi }}\left({\displaystyle \frac{1-2^{2p-\alpha -1}}{{\varphi }^{2p-1}}}+{\displaystyle \frac{2^{2p-\alpha -1}-2^{2p-1}}{{\left(\varphi +\psi \right)}^{2p-1}}}\right)\right)}^{\frac{1}{p}}{\left({\displaystyle \frac{2^{s+1}-1}{2^{s+1}}}{\left|{\mathbb{X}}^{\prime }\left(\varphi ,·\right)\right|}^q+{\displaystyle \frac{1}{2^{s+1}}}{\left|{\mathbb{X}}^{\prime }\left(\psi ,·\right)\right|}^q\right)}^{\frac{1}{q}}\right)\hfill \\ \hfill =& {\displaystyle \frac{\varphi \psi \left(\psi -\varphi \right)}{2{\left(s+1\right)}^{\frac{1}{q}}}}\left(\left({\left({\displaystyle \frac{{}_2\mathit{F}_1\left(2p,1,\alpha p+2;\frac{\psi -\varphi }{\varphi +\psi }\right)}{2^{\alpha p-p+1}{\left(\varphi +\psi \right)}^{2p}\left(\alpha p+1\right)}}\right)}^{\frac{1}{p}}+{\left({\displaystyle \frac{1-2^{2p-\alpha -1}}{\left(\psi -\varphi \right){\varphi }^{2p-1}}}+{\displaystyle \frac{2^{2p-\alpha -1}-2^{2p-1}}{\left(\psi -\varphi \right){\left(\varphi +\psi \right)}^{2p-1}}}\right)}^{\frac{1}{p}}\right)\right.\hfill \\ \hfill & \times {\left({\displaystyle \frac{2^{s+1}-1}{2^{s+1}}}{\left|{\mathbb{X}}^{\prime }\left(\varphi ,·\right)\right|}^q+{\displaystyle \frac{1}{2^{s+1}}}{\left|{\mathbb{X}}^{\prime }\left(\psi ,·\right)\right|}^q\right)}^{\frac{1}{q}}\hfill \\ \hfill & +\left({\left({\displaystyle \frac{{}_2\mathit{F}_1\left(2p,\alpha p+1,\alpha p+2;\frac{\psi -\varphi }{2\psi }\right)}{2^{\alpha p+1}{\psi }^{2p}\left(\alpha p+1\right)}}\right)}^{\frac{1}{p}}+{\left({\displaystyle \frac{2^{2p-1}{\psi }^{2p-1}-{\left(\varphi +\psi \right)}^{2p-1}}{\left(2p-1\right)\left(\psi -\varphi \right){\psi }^{2p-1}{\left(\varphi +\psi \right)}^{2p-1}}}-{\displaystyle \frac{F_1\left(1,-\alpha ,2p,2,\frac{1}{2},\frac{\psi -\varphi }{2\psi }\right)}{2{\psi }^{2p}}}\right)}^{\frac{1}{p}}\right)\hfill \\ \hfill & \times \left.{\left({\displaystyle \frac{1}{2^{s+1}}}{\left|{\mathbb{X}}^{\prime }\left(\varphi ,·\right)\right|}^q+{\displaystyle \frac{2^{s+1}-1}{2^{s+1}}}{\left|{\mathbb{X}}^{\prime }\left(\psi \right)\right|}^q\right)}^{\frac{1}{q}}\right),\hfill \end{array}$$

where we have used

$$\underset{0}{\stackrel{\frac{1}{2}}{\int }}{\displaystyle \frac{{\nu }^{\alpha p}}{{\left(\left(1-\nu \right)\varphi +\nu \psi \right)}^{2p}}}\mathrm{d}\nu ={\displaystyle \frac{2^p}{2^{\alpha p+1}{\left(\varphi +\psi \right)}^{2p}}}\underset{0}{\stackrel{1}{\int }}{\displaystyle \frac{{\left(1-\nu \right)}^{\alpha p}}{{\left(1-\nu \frac{\psi -\varphi }{\varphi +\psi }\right)}^{2p}}}\mathrm{d}\nu ={\displaystyle \frac{{}_2\mathit{F}_1\left(2p,1,\alpha p+2;\frac{\psi -\varphi }{\varphi +\psi }\right)}{2^{\alpha p-p+1}{\left(\varphi +\psi \right)}^{2p}\left(\alpha p+1\right)}},$$

(19)

$$\underset{0}{\stackrel{\frac{1}{2}}{\int }}{\displaystyle \frac{{\nu }^{\alpha p}}{{\left(\nu \varphi +\left(1-\nu \right)\psi \right)}^{2p}}}\mathrm{d}\nu ={\displaystyle \frac{1}{2^{\alpha p+1}{\psi }^{2p}}}\underset{0}{\stackrel{1}{\int }}{\displaystyle \frac{{\nu }^{\alpha p}}{{\left(1-\nu \frac{\psi -\varphi }{2\psi }\right)}^{2p}}}\mathrm{d}\nu ={\displaystyle \frac{{}_2\mathit{F}_1\left(2p,\alpha p+1,\alpha p+2;\frac{\psi -\varphi }{2\psi }\right)}{2^{\alpha p+1}{\psi }^{2p}\left(\alpha p+1\right)}},$$

(20)

$$\begin{array}{cc}\hfill \underset{\frac{1}{2}}{\stackrel{1}{\int }}{\displaystyle \frac{{\left(1-{\nu }^{\alpha }\right)}^p}{{\left(\left(1-\nu \right)\varphi +\nu \psi \right)}^{2p}}}\mathrm{d}\nu \le & \underset{\frac{1}{2}}{\stackrel{1}{\int }}{\displaystyle \frac{1-{\nu }^{\alpha }}{{\left(\left(1-\nu \right)\varphi +\nu \psi \right)}^{2p}}}\mathrm{d}\nu \hfill \\ \hfill =& \underset{0}{\stackrel{\frac{1}{2}}{\int }}{\displaystyle \frac{1-{\left(1-\nu \right)}^{\alpha }}{{\left(\psi -\left(\psi -\varphi \right)\nu \right)}^{2p}}}\mathrm{d}\nu \hfill \\ \hfill =& \underset{0}{\stackrel{\frac{1}{2}}{\int }}{\displaystyle \frac{1}{{\left(\psi -\left(\psi -\varphi \right)\nu \right)}^{2p}}}\mathrm{d}\nu -{\displaystyle \frac{1}{2{\psi }^{2p}}}\underset{0}{\stackrel{1}{\int }}{\left(1-{\displaystyle \frac{1}{2}}t\right)}^{\alpha }{\left(1-{\displaystyle \frac{\psi -\varphi }{2\psi }}\nu \right)}^{-2p}\mathrm{d}\nu \hfill \\ \hfill =& {\displaystyle \frac{2^{2p-1}{\psi }^{2p-1}-{\left(\varphi +\psi \right)}^{2p-1}}{\left(2p-1\right)\left(\psi -\varphi \right){\psi }^{2p-1}{\left(\varphi +\psi \right)}^{2p-1}}}-{\displaystyle \frac{F_1\left(1,-\alpha ,2p,2,\frac{1}{2},\frac{\psi -\varphi }{2\psi }\right)}{2{\psi }^{2p}}}\hfill \end{array}$$

(21)

and

$$\begin{array}{cc}\hfill \underset{\frac{1}{2}}{\stackrel{1}{\int }}{\displaystyle \frac{{\left(1-{\nu }^{\alpha }\right)}^p}{{\left(\nu \varphi +\left(1-\nu \right)\psi \right)}^{2p}}}\mathrm{d}\nu \le & \underset{\frac{1}{2}}{\stackrel{1}{\int }}{\displaystyle \frac{1-{\nu }^{\alpha }}{{\left(\nu \varphi +\left(1-\nu \right)\psi \right)}^{2p}}}\mathrm{d}\nu =\underset{\frac{1}{2}}{\stackrel{1}{\int }}{\displaystyle \frac{1-{\nu }^{\alpha }}{{\left(\psi -\left(\psi -\varphi \right)\nu \right)}^{2p}}}\mathrm{d}\nu \hfill \\ \hfill =& \underset{\frac{1}{2}}{\stackrel{1}{\int }}{\displaystyle \frac{1}{{\left(\psi -\left(\psi -\varphi \right)\nu \right)}^{2p}}}\mathrm{d}\nu -\underset{\frac{1}{2}}{\stackrel{1}{\int }}{\displaystyle \frac{{\nu }^{\alpha }}{{\left(\psi -\left(\psi -\varphi \right)\nu \right)}^{2p}}}\mathrm{d}\nu \left[{\displaystyle \frac{1}{2}}\underset{0}{\stackrel{1}{\int }}{\displaystyle \frac{{\left(1-\frac{1}{2}\nu \right)}^{\alpha }}{{\left(\varphi +\left(\frac{\psi -\varphi }{2}\right)\nu \right)}^{2p}}}\mathrm{d}\nu \right]\hfill \\ \hfill =& \underset{\frac{1}{2}}{\stackrel{1}{\int }}{\displaystyle \frac{1}{{\left(\psi -\left(\psi -\varphi \right)\nu \right)}^{2p}}}\mathrm{d}\nu -2^{2p-\alpha -1}\underset{0}{\stackrel{1}{\int }}{\displaystyle \frac{{\left(1+\nu \right)}^{\alpha }}{{\left(\left(\varphi +\psi \right)-\left(\psi -\varphi \right)\nu \right)}^{2p}}}\mathrm{d}\nu \hfill \\ \hfill \le & {\displaystyle \frac{1}{\psi -\varphi }}\left({\displaystyle \frac{1-2^{2p-\alpha -1}}{{\varphi }^{2p-1}}}+{\displaystyle \frac{2^{2p-\alpha -1}-2^{2p-1}}{{\left(\varphi +\psi \right)}^{2p-1}}}\right).\hfill \end{array}$$

(22)

The proof is completed. □

Theorem 7.

Let $0<\varphi <\psi$ and $\mathbb{X}:[\varphi ,\psi ]\times \mathrm{\Lambda }\to \mathbb{R}$ be a mean-square differentiable stochastic process on ${\mathcal{D}}^{\circ }$. If ${\left|{\mathbb{X}}^{\prime }\right|}^q$ is a harmonically s-convex stochastic process on $\left[\varphi ,\psi \right]$ for $s\in (0,1]$ and $q>0$, then for $\alpha >0$ we have

$$\begin{array}{cc}\hfill & \left|\mathbb{X}\left({\displaystyle \frac{2\varphi \psi }{\varphi +\psi }},·\right)-{\displaystyle \frac{\mathrm{\Gamma }\left(\alpha +1\right)}{2}}{\left({\displaystyle \frac{\varphi \psi }{\psi -\varphi }}\right)}^{\alpha }\left({\mathbf{J}}_{{\frac{1}{\varphi }}^{-}}^{\alpha }\left(\mathbb{X}\circ \mathrm{h}\right)\left({\displaystyle \frac{1}{\psi }},·\right)+{\mathbf{J}}_{{\frac{1}{\psi }}^{+}}^{\alpha }\left(\mathbb{X}\circ \mathrm{h}\right)\left({\displaystyle \frac{1}{\varphi }}\right)\right)\right|\hfill \\ \hfill \le & {\displaystyle \frac{\varphi \psi \left(\psi -\varphi \right)}{2}}\left({\left({\displaystyle \frac{{}_2\mathit{F}_1\left(2,1,\alpha +2;\frac{\psi -\varphi }{\varphi +\psi }\right)}{2^{\alpha -2}{\left(\varphi +\psi \right)}^2\left(\alpha +1\right)}}\right)}^{1-\frac{1}{q}}\right.{\left({\mathcal{M}}_1(s,\alpha ){\left|{\mathbb{X}}^{\prime }\left(\varphi ,·\right)\right|}^q+{\mathcal{M}}_2(s,\alpha ){\left|{\mathbb{X}}^{\prime }\left(\psi ,·\right)\right|}^q\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left({\displaystyle \frac{{}_2\mathit{F}_1\left(2,\alpha +1,\alpha +2;\frac{\psi -\varphi }{2\psi }\right)}{2^{\alpha +1}{\psi }^2\left(\alpha +1\right)}}\right)}^{1-\frac{1}{q}}{\left({\mathcal{M}}_3(s,\alpha ){\left|{\mathbb{X}}^{\prime }\left(\varphi ,·\right)\right|}^q+{\mathcal{M}}_4(s,\alpha ){\left|{\mathbb{X}}^{\prime }\left(\psi ,·\right)\right|}^q\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left({\displaystyle \frac{1}{\psi \left(\varphi +\psi \right)}}-{\displaystyle \frac{F_1\left(1,-\alpha ,2,2,\frac{1}{2},\frac{\psi -\varphi }{2\psi }\right)}{2{\psi }^2}}\right)}^{1-\frac{1}{q}}{\left({\mathcal{M}}_5(s,\alpha ){\left|{\mathbb{X}}^{\prime }\left(\varphi ,·\right)\right|}^q+{\mathcal{M}}_6(s,\alpha ){\left|{\mathbb{X}}^{\prime }\left(\psi ,·\right)\right|}^q\right)}^{\frac{1}{q}}\hfill \\ \hfill & \left.+{\left({\displaystyle \frac{1}{\psi -\varphi }}\left({\displaystyle \frac{1-2^{2-\alpha }}{\varphi }}+{\displaystyle \frac{2^{2-\alpha }-2}{\left(\varphi +\psi \right)}}\right)\right)}^{1-\frac{1}{q}}{\left({\mathcal{M}}_7(s,\alpha ){\left|{\mathbb{X}}^{\prime }\left(\varphi ,·\right)\right|}^q+{\mathcal{M}}_8(s,\alpha ){\left|{\mathbb{X}}^{\prime }\left(\psi ,·\right)\right|}^q\right)}^{\frac{1}{q}}\right),\hfill \end{array}$$

where ${\mathcal{M}}_k(s,\alpha )$ ($k=1,\dots ,8$) are defined as in (1)–(8), respectively.

Proof. 

From Lemma 3, absolute value, power mean inequality and harmonically s-convexity of ${\left|{\mathbb{X}}^{\prime }\right|}^q$, we have

$$\begin{array}{cc}\hfill & \left|\mathbb{X}\left({\displaystyle \frac{2\varphi \psi }{\varphi +\psi }},·\right)-{\displaystyle \frac{\mathrm{\Gamma }\left(\alpha +1\right)}{2}}{\left({\displaystyle \frac{\varphi \psi }{\psi -\varphi }}\right)}^{\alpha }\left({\mathbf{J}}_{{\frac{1}{\varphi }}^{-}}^{\alpha }\left(\mathbb{X}\circ \mathrm{h}\right)\left({\displaystyle \frac{1}{\psi }},·\right)+{\mathbf{J}}_{{\frac{1}{\psi }}^{+}}^{\alpha }\left(\mathbb{X}\circ \mathrm{h}\right)\left({\displaystyle \frac{1}{\varphi }},·\right)\right)\right|\hfill \\ \hfill \le & {\displaystyle \frac{\varphi \psi \left(\psi -\varphi \right)}{2}}\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}{\displaystyle \frac{{\nu }^{\alpha }}{{\left(\left(1-\nu \right)\varphi +\nu \psi \right)}^2}}\left|{\mathbb{X}}^{\prime }\left({\displaystyle \frac{\varphi \psi }{\left(1-\nu \right)\varphi +\nu \psi }},·\right)\right|+{\displaystyle \frac{{\nu }^{\alpha }}{{\left(\nu \varphi +(1-\nu )\psi \right)}^2}}\left|{\mathbb{X}}^{\prime }\left({\displaystyle \frac{\varphi \psi }{\nu \varphi +\left(1-\nu \right)\psi }},·\right)\right|\mathrm{d}\nu \right.\hfill \\ \hfill & +\left.\underset{\frac{1}{2}}{\stackrel{1}{\int }}{\displaystyle \frac{1-{\nu }^{\alpha }}{{\left(\left(1-\nu \right)\varphi +\nu \psi \right)}^2}}\left|{\mathbb{X}}^{\prime }\left({\displaystyle \frac{\varphi \psi }{\left(1-\nu \right)\varphi +\nu \psi }},·\right)\right|+{\displaystyle \frac{{\nu }^{\alpha }}{{\left(\nu \varphi +(1-\nu )\psi \right)}^2}}\left|{\mathbb{X}}^{\prime }\left({\displaystyle \frac{\varphi \psi }{\nu \varphi +\left(1-\nu \right)\psi }},·\right)\right|\mathrm{d}\nu \right)\hfill \\ \hfill \le & {\displaystyle \frac{\varphi \psi \left(\psi -\varphi \right)}{2}}\left({\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}{\displaystyle \frac{{\nu }^{\alpha }}{{\left(\left(1-\nu \right)\varphi +\nu \psi \right)}^2}}\mathrm{d}\nu \right)}^{1-\frac{1}{q}}{\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}{\displaystyle \frac{{\nu }^{\alpha }}{{\left(\left(1-\nu \right)\varphi +\nu \psi \right)}^2}}{\left|{\mathbb{X}}^{\prime }\left({\displaystyle \frac{\varphi \psi }{\left(1-\nu \right)\varphi +\nu \psi }},·\right)\right|}^q\mathrm{d}\nu \right)}^{\frac{1}{q}}\right.\hfill \\ \hfill & +{\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}{\displaystyle \frac{{\nu }^{\alpha }}{{\left(\nu \varphi +(1-\nu )\psi \right)}^2}}\mathrm{d}\nu \right)}^{1-\frac{1}{q}}{\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}{\displaystyle \frac{{\nu }^{\alpha }}{{\left(\nu \varphi +\left(1-\nu \right)\psi \right)}^2}}{\left|{\mathbb{X}}^{\prime }\left({\displaystyle \frac{\varphi \psi }{\nu \varphi +\left(1-\nu \right)\psi }},·\right)\right|}^q\mathrm{d}\nu \right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(\underset{\frac{1}{2}}{\stackrel{1}{\int }}{\displaystyle \frac{1-{\nu }^{\alpha }}{{\left(\left(1-\nu \right)\varphi +\nu \psi \right)}^2}}\mathrm{d}\nu \right)}^{1-\frac{1}{q}}{\left(\underset{\frac{1}{2}}{\stackrel{1}{\int }}{\displaystyle \frac{1-{\nu }^{\alpha }}{{\left(\left(1-\nu \right)\varphi +\nu \psi \right)}^2}}{\left|{\mathbb{X}}^{\prime }\left({\displaystyle \frac{\varphi \psi }{\left(1-\nu \right)\varphi +\nu \psi }},·\right)\right|}^q\mathrm{d}\nu \right)}^{\frac{1}{q}}\hfill \\ \hfill & \left.+{\left(\underset{\frac{1}{2}}{\stackrel{1}{\int }}{\displaystyle \frac{1-{\nu }^{\alpha }}{{\left(\nu \varphi +\left(1-\nu \right)\psi \right)}^2}}\mathrm{d}\nu \right)}^{1-\frac{1}{q}}{\left(\underset{\frac{1}{2}}{\stackrel{1}{\int }}{\displaystyle \frac{1-{\nu }^{\alpha }}{{\left(\nu \varphi +\left(1-\nu \right)\psi \right)}^2}}{\left|{\mathbb{X}}^{\prime }\left({\displaystyle \frac{\varphi \psi }{\nu \varphi +\left(1-\nu \right)\psi }},·\right)\right|}^q\mathrm{d}\nu \right)}^{\frac{1}{q}}\right)\hfill \\ \hfill \le & {\displaystyle \frac{\varphi \psi \left(\psi -\varphi \right)}{2}}\left({\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}{\displaystyle \frac{{\nu }^{\alpha }}{{\left(\left(1-\nu \right)\varphi +\nu \psi \right)}^2}}\mathrm{d}\nu \right)}^{1-\frac{1}{q}}{\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}{\displaystyle \frac{{\nu }^{\alpha }}{{\left(\left(1-\nu \right)\varphi +\nu \psi \right)}^2}}\left({\nu }^s{\left|{\mathbb{X}}^{\prime }\left(\varphi ,·\right)\right|}^q+{\left(1-\nu \right)}^s{\left|{\mathbb{X}}^{\prime }\left(\psi ,·\right)\right|}^q\right)\mathrm{d}\nu \right)}^{\frac{1}{q}}\right.\hfill \\ \hfill & +{\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}{\displaystyle \frac{{\nu }^{\alpha }}{{\left(\nu \varphi +\left(1-\nu \right)\psi \right)}^2}}\mathrm{d}\nu \right)}^{1-\frac{1}{q}}{\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}{\displaystyle \frac{{\nu }^{\alpha }}{{\left(\nu \varphi +\left(1-\nu \right)\psi \right)}^2}}\left({\left(1-\nu \right)}^s{\left|{\mathbb{X}}^{\prime }\left(\varphi ,·\right)\right|}^q+{\nu }^s{\left|{\mathbb{X}}^{\prime }\left(\psi ,·\right)\right|}^q\right)\mathrm{d}\nu \right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(\underset{\frac{1}{2}}{\stackrel{1}{\int }}{\displaystyle \frac{1-{\nu }^{\alpha }}{{\left(\left(1-\nu \right)\varphi +\nu \psi \right)}^2}}\mathrm{d}\nu \right)}^{1-\frac{1}{q}}{\left(\underset{\frac{1}{2}}{\stackrel{1}{\int }}{\displaystyle \frac{1-{\nu }^{\alpha }}{{\left(\left(1-\nu \right)\varphi +\nu \psi \right)}^2}}\left({\nu }^s{\left|{\mathbb{X}}^{\prime }\left(\varphi ,·\right)\right|}^q+{\left(1-\nu \right)}^s{\left|{\mathbb{X}}^{\prime }\left(\psi ,·\right)\right|}^q\right)\mathrm{d}\nu \right)}^{\frac{1}{q}}\hfill \\ \hfill & +\left.{\left(\underset{\frac{1}{2}}{\stackrel{1}{\int }}{\displaystyle \frac{1-{\nu }^{\alpha }}{{\left(\nu \varphi +(1-\nu )\psi \right)}^2}}\mathrm{d}\nu \right)}^{1-\frac{1}{q}}{\left(\underset{\frac{1}{2}}{\stackrel{1}{\int }}{\displaystyle \frac{1-{\nu }^{\alpha }}{{\left(\nu \varphi +\left(1-\nu \right)\psi \right)}^2}}\left({\left(1-\nu \right)}^s{\left|{\mathbb{X}}^{\prime }\left(\varphi ,·\right)\right|}^q+{\nu }^s{\left|{\mathbb{X}}^{\prime }\left(\psi ,·\right)\right|}^q\right)\mathrm{d}\nu \right)}^{\frac{1}{q}}\right)\hfill \\ \hfill \le & {\displaystyle \frac{\varphi \psi \left(\psi -\varphi \right)}{2}}\left({\left({\displaystyle \frac{{}_2\mathit{F}_1\left(2,1,\alpha +2;\frac{\psi -\varphi }{\varphi +\psi }\right)}{2^{\alpha -2}{\left(\varphi +\psi \right)}^2\left(\alpha +1\right)}}\right)}^{1-\frac{1}{q}}\right.{\left({\mathcal{M}}_1(s,\alpha ){\left|{\mathbb{X}}^{\prime }\left(\varphi ,·\right)\right|}^q+{\mathcal{M}}_2(s,\alpha ){\left|{\mathbb{X}}^{\prime }\left(\psi ,·\right)\right|}^q\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left({\displaystyle \frac{{}_2\mathit{F}_1\left(2,\alpha +1,\alpha +2;\frac{\psi -\varphi }{2\psi }\right)}{2^{\alpha +1}{\psi }^2\left(\alpha +1\right)}}\right)}^{1-\frac{1}{q}}{\left({\mathcal{M}}_3(s,\alpha ){\left|{\mathbb{X}}^{\prime }\left(\varphi ,·\right)\right|}^q+{\mathcal{M}}_4(s,\alpha ){\left|{\mathbb{X}}^{\prime }\left(\psi ,·\right)\right|}^q\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left({\displaystyle \frac{1}{\psi \left(\varphi +\psi \right)}}-{\displaystyle \frac{F_1\left(1,-\alpha ,2,2,\frac{1}{2},\frac{\psi -\varphi }{2\psi }\right)}{2{\psi }^2}}\right)}^{1-\frac{1}{q}}{\left({\mathcal{M}}_5(s,\alpha ){\left|{\mathbb{X}}^{\prime }\left(\varphi ,·\right)\right|}^q+{\mathcal{M}}_6(s,\alpha ){\left|{\mathbb{X}}^{\prime }\left(\psi ,·\right)\right|}^q\right)}^{\frac{1}{q}}\hfill \\ \hfill & \left.+{\left({\displaystyle \frac{1}{\psi -\varphi }}\left({\displaystyle \frac{1-2^{2-\alpha }}{\varphi }}+{\displaystyle \frac{2^{2-\alpha }-2}{\left(\varphi +\psi \right)}}\right)\right)}^{1-\frac{1}{q}}{\left({\mathcal{M}}_7(s,\alpha ){\left|{\mathbb{X}}^{\prime }\left(\varphi ,·\right)\right|}^q+{\mathcal{M}}_8(s,\alpha ){\left|{\mathbb{X}}^{\prime }\left(\psi ,·\right)\right|}^q\right)}^{\frac{1}{q}}\right),\hfill \end{array}$$

where we have used (19)–(22).

The proof is completed. □

6. Illustrative Example and Application

To confirm the validity of our results and highlight their applicability, we first present a numerical example with graphical illustration that supports our findings, followed by an application to special means.

6.1. Numerical Example and Graphical Illustration

Example 1.

Let $\varphi =1$, $\psi =2$, and define the function

$$\mathbb{X}(u,.)={\displaystyle \frac{u^{1+s}}{1+s}},u\in [1,2],$$

where $s\in (0,1]$. Its derivative is ${\mathbb{X}}^{\prime }(u,.)=u^s$, which is known to be harmonically s-convex on $(0,\infty )$; see [9].

Using symbolic and numerical computation, we evaluate both sides of the inequality obtained in Theorem 5 over the domain $\alpha \in (0,10]$ and $s\in (0,1]$. The resulting surfaces, depicted in Figure 1, confirm the validity of the inequality for all sampled parameter values, thereby numerically verifying the sharpness and correctness of our result.

6.2. Application to Special Means

The inequalities developed in the preceding section are now applied to special means, demonstrating their utility in related problems.

For arbitrary real numbers $\varphi ,\psi$ we have the following:

The Harmonic mean: $\mathcal{H}\left(\varphi ,\psi \right)={\displaystyle \frac{2\varphi \psi }{\varphi +\psi }}$.

The Geometric mean: $\mathcal{G}\left(\varphi ,\psi \right)=\sqrt{\varphi \psi }$.

The Logarithmic mean: $\mathcal{L}\left(\varphi ,\psi \right)={\displaystyle \frac{\psi -\varphi }{ln\psi -ln\varphi }}$, $\varphi ,\psi >0,\varphi \ne \psi$.

Proposition 1.

Let $\varphi ,\psi \in \mathbb{R}$ with $0<\varphi <\psi$, then we have

$$\begin{array}{cc}\hfill & \left|\mathcal{H}\left(\varphi ,\psi \right)\left(1-ln\mathcal{H}\left(\varphi ,\psi \right)\right)+\left(1-ln\mathcal{G}\left(\varphi ,\psi \right)\right){\mathcal{G}}^2\left(\varphi ,\psi \right){\mathcal{L}}^{-1}\left(\varphi ,\psi \right)\right|\hfill \\ \hfill \le & {\displaystyle \frac{\varphi \psi \left(\psi -\varphi \right)}{2}}\left(\left({\displaystyle \frac{2(\varphi +\psi )}{{\left(\psi -\varphi \right)}^3}}ln{\displaystyle \frac{2\psi }{\varphi +\psi }}+{\displaystyle \frac{4\varphi }{{\left(\psi -\varphi \right)}^3}}ln{\displaystyle \frac{2\varphi }{\varphi +\psi }}\right)\left|ln\left(\varphi \right)\right|\right.\hfill \\ \hfill & +\left.\left({\displaystyle \frac{2(\varphi +\psi )}{{\left(\psi -\varphi \right)}^3}}ln{\displaystyle \frac{\varphi +\psi }{2\psi }}+{\displaystyle \frac{4\psi }{{\left(\psi -\varphi \right)}^3}}ln{\displaystyle \frac{\varphi +\psi }{2\psi }}\right)\left|{ln}^{\prime }\left(\psi \right)\right|\right).\hfill \end{array}$$

Proof. 

The assertion follows from Corollary 4, applied to the function

$$\mathbb{X}(u,·)=ulnu-u,$$

whose derivative satisfies $\left|{\mathbb{X}}^{\prime }(u,·)\right|=lnu$, a harmonically convex function. □

7. Conclusions

In this paper, we have introduced the concept of harmonically s-convex stochastic processes and presented the associated Hermite–Hadamard inequality via stochastic Riemann–Liouville fractional integrals. Subsequently, we introduced a new stochastic fractional integral identity and used it to establish a comprehensive set of midpoint-type inequalities tailored to mean-square differentiable, harmonically s-convex stochastic processes. Our results not only unify and generalize several known inequalities but also provide refined error bounds that are particularly relevant in the analysis of fractional numerical integration methods. These improved bounds, which account for stochastic behavior and harmonic s-convexity, are particularly relevant in the error analysis of numerical methods for fractional integrals, where sharp control of approximation errors is essential for schemes involving memory effects or non-local dynamics.