On strongly generalized convex stochastic processes

Abstract

In this paper, we introduce the notion of strongly generalized convex functions which is called as strongly η-convex stochastic processes. We prove the Hermite-Hadamard, Ostrowski type inequality, and obtain some important inequalities for above processes. Some previous results are special cases of the results obtained in this paper.

Keywords:

• Convex stochastic processes
• strongly convex stochastic processes
• Hermite-Hadamard type integral inequality
• Hölder’s inequality

MATHEMATICS SUBJECT CLASSIFICATION (2010):

• 26A51
• 26D15
• 52A01
• 60G99

1. Introduction

Karamardian (1969) introduced strongly convex functions. However, we can find some references (Merentes and Nikodem 2010; Nikodem and Pales 2011) citing Polyak (1966) as being the pioneer to introduce this notion. It is well known that every continuously differentiable function is strongly monotone if its Jacobian matrix is strongly positive definite (Karamardian 1969). Awan et al. (2017) introduced the notion of strongly η-convex functions and obtained some new integral inequalities of Hermite-Hadamard and Hermite-Hadamard-Fejér type for strongly η-convex functions.

Practical applications of stochastic processes include areas of image processing, signal processing, control theory, information theory, computer science, cryptography, telecommunications, neuroscience, statistics, physics, biology, chemistry and ecology, see (Allen 2010; Kotrys 2012; Nikodem 1980; Sobczyk 2013) and their references. Nikodem (1980) introduced some powerful properties of convex stochastic processes. Nikodem (1980) proved that a convex stochastic processes $X:I\times \Omega \to \overline{\mathbb{R}}$ is continuous if and only if for all $u,v\in I$ and $\lambda \in [0,1]$ $$X(\lambda u+(1-\lambda )v,.)\le \lambda X(u,.)+(1-\lambda )X(v,.)\mathrm{ }\text{almost}\mathrm{ }\text{everywhere},$$ where $\overline{\mathbb{R}}$ denotes the extended real line. It is well known that if a stochastic process $X:I\times \Omega \to \overline{\mathbb{R}}$ is convex and measurable, then it is continuous (Nikodem 1980). Kuhn (2006) studied the convex stochastic programs with a generalized non convex dependence on the random parameters. Kuhn (2006) proved that, under certain conditions, the saddle structure can be restored by adding specific random variables to the profit functions. These random variables are referred to as ‘correction terms’. Allen (2010) introduced basic theory of stochastic processes and applying these methods to biological problems, such as enzyme kinetics, population extinction, the spread of epidemics and the genetics of inbreeding.

Kotrys (2013) defined the strongly convex stochastic processes and derived the Hermite-Hadamard inequality, Jensen inequality, Kuhn and Bernstein theorem. A stochastic process $X:I\times \Omega \to \mathbb{R}$ is strongly $\lambda$-convex with modulus $\mu (.)$ if and only if the stochastic process $Y:I\times \Omega \to \mathbb{R}$ defined by $Y(u,.)=X(u,.)-\mu (.)u^2$ is $\lambda$-convex (Kotrys 2013). Further, Ibrahim (2020) introduced the concept of strongly h-convex stochastic process. Jung et al. (2021) introduced the notion of η-convex stochastic processes and established Jensen, Hermite-Hadamard and Ostrowski type inequalities. Further, Fu et al. (2021) derived Hölder-İşcan and Improved power mean integral inequalities, and proved Hermite-Hadamard type integral inequalities for $n-$ polynomial stochastic processes. For more study of Hermite-Hadamard inequalities, we refer to (Karahan and Okur 2018; Omaba and Nwaeze 2020; Özcan 2019; Sharma, Mishra, and Hamdi 2020, Sharma et al. 2021).

Recently, Farid et al. (2021) proposed refinements of fractional versions of the Hadamard inequalities for Caputo fractional derivatives using strongly convex functions. Tariq et al. (2021) defined harmonic s-type convex functions and investigated the refinements of Ostrowski type inequalities for s-type convex functions. Further, Omaba, Omenyi and Omenyi (2021) obtained the new Hadamard type inequalities for a class of s-Godunova-Levin functions of the second kind for fractional integrals. Iftikhar et al. (2021) introduced some new generalized different types of kernels, development of new identities and new error bounds of Ostrowski type inequalities for first and second derivable mappings.

Stochastic convexity and its applications are very important in mathematics, probability, so researchers are attracting to develop generalized stochastic convexity. However, Hermite-Hadamard and some other integral inequalities for strongly η-convex stochastic processes have not been studied. So we propose a new class of convex functions known as strongly η-convex stochastic processes. The definition of strongly η-convex stochastic processes is motivated by the definition of η-convex, strongly convex stochastic processes and strongly η-convex functions. Thus, the purpose of this paper is to establish Hermite-Hadamard, Ostrowski and some other integral inequalities for strongly η-convex stochastic processes.

The organization of the paper is as follows: In Section 2, we recall some basic results that are necessary for our main results. In Section 3, we introduce the concept of strongly η-convex stochastic processes. Further, we prove the Hermite-Hadamard inequality, Ostrowski inequality and some other interesting inequalities for this paper. Some special cases of these results are also investigated in Section 3. In Section 4, we discuss the conclusions and future directions of this study.

2. Preliminaries

A stochastic process X(t) is a function which maps the index set T into the space S of random variable defined on $(\Omega ,A,P).$

Definition 1.

(Sobczyk 2013) The stochastic process $X:I\times \Omega \to \mathbb{R}$ is called

• continuous in probability in interval I, if for all $u\in I$ $$\begin{array}{c}P-\underset{u\to u_0}{\mathrm{\text{lim}}}X(u,.)=X(u_0,.),\hfill \end{array}$$ where $P-\mathrm{\text{lim}}$ denotes the limit in probability;

• mean square continuous in the interval I, if for all $u_0\in I$ $$\begin{array}{c}P-\underset{u\to u_0}{\mathrm{\text{lim}}}E[X(u,.)-X(u_0,.)]=0,\hfill \end{array}$$ where $E[X(u,.)]$ denotes the expectation value of the random variable $X(u,.);$

• increasing(decreasing), if for all $u,v\in I$ such that $u<v,$ $$\begin{array}{c}X(u,.)\le X(v,.),\hspace{0.17em}\hspace{0.17em}(X(u,.)\ge X(v,.));\hfill \end{array}$$

• monotonic if it is increasing or decreasing;

• differentiable at a point $u\in I$ if there is a random variable $X\prime (u,.):\Omega \to \mathbb{R}$ $$\begin{array}{c}X\prime (u,.)=P-\underset{u\to u_0}{\mathrm{\text{lim}}}\frac{X(u,.)-X(u_0,.)}{u-u_0}.\hfill \end{array}$$

Definition 2.

(Sobczyk 2013) Suppose that $X:I\times \Omega \to \mathbb{R}$ is a stochastic process with $E[X{(u)}^2]<\infty$ for all $u\in I$ and $[c,d]\in I,\hspace{0.17em}c=u_0<u_1<u_2<\dots <u_n=d$ is a partition of $[c,d]$ and ${\Theta }_k\in [u_{k-1},u_k]$ for all $k=1,2,\dots n.$ Further, suppose that $Y:I\times \Omega \to \mathbb{R}$ be a random variable. Then, it is said to be mean-square integral of the process X on $[c,d],$ if for each normal sequence of partitions of the interval $[c,d]$ and for each ${\Theta }_k\in [u_{k-1},u_k],\hspace{0.17em}k=1,2,\dots n,$ we have $$\begin{array}{c}\underset{n\to \infty }{\mathrm{\text{lim}}}E\left[{\left(\sum _{k=1}^{n}X({\Theta }_k)\lambda (u_k-u_{k-1})-Y\right)}^2\right]=0.\hfill \end{array}$$

Then, we can write $$\begin{array}{c}Y(.)={\int }_c^{d}X(v,.)\mathit{\text{dv}}\hspace{0.17em}\hspace{0.17em}(a.e.).\hfill \end{array}$$

Also, mean square integral operator is increasing, that is, $$\begin{array}{c}{\int }_c^{d}X(u,.)\mathit{\text{du}}\le {\int }_c^{d}Y(u,.)\mathit{\text{du}}\hspace{0.17em}\hspace{0.17em}(a.e.),\hfill \end{array}$$ where $X(u,.)\le Y(u,.)\text{in}\hspace{0.17em}[c,d].$

Theorem 1.

(Kotrys 2012) Let $X:I\times \Omega \to \mathbb{R}$ is convex and mean square continuous in the interval I. Then for any $c,d\in I,$ we have $$\begin{array}{c}X\left(\frac{c+d}{2},.\right)\le \frac{1}{d-c}{\int }_c^{d}X(u,.)\mathit{\text{du}}\le \frac{X(c,.)+X(d,.)}{2}\hspace{0.17em}\hspace{0.17em}(a.e.).\hfill \end{array}$$

Definition 3.

(Kotrys 2013) Let $\mu :\Omega \to \mathbb{R}$ denote a positive random variable. The stochastic process $X:I\times \Omega \to \mathbb{R}$ is called strongly convex with modulus $\mu (.)>0,$ if for all $\lambda \in [0,1]$ and $u,v\in I$ the inequality $X(\lambda u+(1-\lambda )v,.)\le \lambda X(u,.)+(1-\lambda )X(v,.)-\mu (.)\lambda (1-\lambda ){(u-v)}^2\hspace{0.17em}(a.e.)$ is satisfied.

Theorem 2.

(Kotrys 2013) Let $X:I\times \Omega \to \mathbb{R}$ be a stochastic process, which is strongly Jensen convex with modulus $\mu (.)$ and mean square continuous in the interval I. Then for any $c,d\in I,$ we have $$\begin{array}{c}X\left(\frac{c+d}{2},.\right)+\frac{\mu (.)}{12}{(d-c)}^2\le \frac{1}{d-c}{\int }_c^{d}X(u,.)\mathit{\text{du}}\le \frac{X(c,.)+X(d,.)}{2}-\frac{\mu (.)}{6}{(d-c)}^2\hspace{0.17em}\hspace{0.17em}(a.e.).\hfill \end{array}$$

Definition 4.

(Jung et al. 2021) Let $(\Omega ,A,P)$ be a probability space and $I\subseteq \mathbb{R}$ be an interval, then $X:I\times \Omega \to \mathbb{R}$ is an η-convex stochastic process if $$X(\lambda u+(1-\lambda )v,.)\le X(v,.)+\lambda \eta (X(u,.),X(v,.))\hspace{0.17em}(a.e.),\hspace{0.17em}\forall u,v\in I \mathit{\text{and\hspace{0.5em}}}\lambda \in [0,1].$$

Theorem 3.

(Fu et al. 2021; Hölder-İşcan integral inequality) Let $X,Y:[c,d]\times \Omega \to \mathbb{R}$ be real stochastic process and $|X\prime {|}^p,\hspace{0.17em}|Y\prime {|}^p$ be mean square integrable on $[c,d].$ If p > 1 and $\frac{1}{p}+\frac{1}{q}=1,$ then the following inequality holds almost everywhere: $$\begin{array}{cc}{\int }_c^d|X(u,.)Y(u,.)|\mathit{\text{du}}\hfill & \le \frac{1}{d-c}[{({\int }_c^d(d-u)|X(u,.){|}^p\mathit{\text{du}})}^{1/p}{({\int }_c^d(d-u)|Y(u,.){|}^q\mathit{\text{du}})}^{1/q}\hfill \\ \hspace{0.17em}\hfill & +{({\int }_c^d(u-c)|X(u,.){|}^p\mathit{\text{du}})}^{1/p}{({\int }_c^d(u-c)|Y(u,.){|}^q\mathit{\text{du}})}^{1/q}].\hfill \end{array}$$

Theorem 4.

(Fu et al. 2021; Improved power mean integral inequality) Let $X,Y:[c,d]\times \Omega \to \mathbb{R}$ be real stochastic process and $\left|X\right|,\hspace{0.17em}\left|X\right||Y{|}^q$ be mean square integrable on $[c,d].$ If $q\ge 1,$ then the following inequality holds almost everywhere: $$\begin{array}{cc}{\int }_c^d|X(u,.)Y(u,.)|\mathit{\text{du}}\hfill & \le \frac{1}{d-c}[{({\int }_c^d(d-u)|X(u,.)|\mathit{\text{du}})}^{1-\frac{1}{q}}{({\int }_c^d(d-u)|X(u,.)||Y(u,.){|}^q\mathit{\text{du}})}^{1/q}\hfill \\ \hspace{0.17em}\hfill & +{({\int }_c^d(u-c)|X(u,.)|\mathit{\text{du}})}^{1-\frac{1}{q}}{({\int }_c^d(u-c)|X(u,.)||Y(u,.){|}^q\mathit{\text{du}})}^{1/q}].\hfill \end{array}$$

Lemma 1.

(Gonzales, Materano, and Lopez 2016) Let $X:I\times \Omega \to \mathbb{R}$ be a stochastic process which is mean square differentiable on I⁰. If $X\prime$ is mean square integrable on $[c,d],$ where $c,d\in I$ with $c<d,$ then the following equality holds $$\begin{array}{cc}X(t,.)-\frac{1}{d-c}{\int }_c^{d}X(u,.)\mathit{\text{du}}\hfill & =\frac{{(x-c)}^2}{d-c}{\int }_0^{1}tX\prime (\mathit{\text{tx}}+(1-t)c,.)\mathit{\text{dt}}\hfill \\ \hspace{0.17em}\hfill & -\frac{{(d-x)}^2}{d-c}{\int }_0^{1}tX\prime (\mathit{\text{tx}}+(1-t)d,.)\mathit{\text{dt}},\hspace{0.17em}(a.e),\hspace{0.17em}\text{for}\mathrm{ }\text{each}\hspace{0.17em}x\in [c,d].\hfill \end{array}$$

Lemma 2.

(Fu et al. 2021) Let $X:I\times \Omega \to \mathbb{R}$ be a mean square differentiable stochastic process on I⁰ and $X\prime$ is mean square integrable on $[c,d]$, where $c,d\in I,c<d.$ Then we have almost everywhere $$\begin{array}{c}\frac{X(c,.)+X(d,.)}{2}-\frac{1}{d-c}{\int }_c^{d}X(u,.)\mathit{\text{du}}=\frac{d-c}{2}{\int }_0^1(1-2\lambda )X\prime (\lambda c+(1-\lambda )d,.)d\lambda .\hfill \end{array}$$

Theorem 5.

(Jung et al. 2021) Suppose that $X:[c,d]\times \Omega \to \mathbb{R}$ is an η-convex stochastic process such that η is bounded above on $X[c,d]\times X[c,d],$ then the following inequalities hold almost everywhere: $$\begin{array}{cc}\hfill & X\left(\frac{c+d}{2},.\right)-\frac{M_{\eta }}{2}\le \frac{1}{d-c}{\int }_c^{d}X(u,.)\mathit{\text{du}}\hfill \\ \hspace{0.17em}\hfill & \le \frac{X(c,.)+X(d,.)}{2}+\frac{1}{4}(\eta (X(c,.),X(d,.))+\eta (X(d,.),X(c,.)))\hfill \\ \hspace{0.17em}\hfill & \le \frac{X(c,.)+X(d,.)}{2}+M_{\eta }.\hfill \end{array}$$

3. Main results

In this section, we derive our main results.

Definition 5.

Let $(\Omega ,A,P)$ be a probability space and $I\subseteq \mathbb{R}$ be an interval. Let $\mu (.)$ denote a positive random variable, then $X:I\times \Omega \to \mathbb{R}$ is said to be strongly η-convex stochastic process with respect to $\eta :X(I)\times X(I)\to \mathbb{R}$ and modulus $\mu (.)>0$ if $$\begin{array}{cc}\hfill & X(\lambda u+(1-\lambda )v,.)\le X(v,.)+\lambda \eta (X(u,.),X(v,.))-\mu (.)\lambda (1-\lambda ){(v-u)}^2\hspace{0.17em}(a.e.)\hfill \\ \hspace{0.17em}\hspace{0.17em}\hfill & \hspace{0.17em}\forall u,v\in I \mathit{\text{and\hspace{0.5em}}}\lambda \in [0,1].\hfill \end{array}$$

Remark 1.

If $\eta (X(u,.),X(v,.))=X(u,.)-X(v,.)$, then the definition of strongly η-convex stochastic process reduces to the definition of strongly convex stochastic process proposed by Kotrys (2013). When $\mu (.)=0,$ then above definition reduces to the definition of η-convex stochastic process (Jung et al. 2021).

Example 1.

Let $X:(0,\infty )\times \Omega \to \mathbb{R}$ be a stochastic process defined as $X(u,.)=u,\hspace{0.17em}$ and $\eta :X((0,\infty ))\times X((0,\infty ))\to \mathbb{R},\eta (X(u,.),X(v,.))={(X(u,.)-X(v,.))}^2+X(u,.)+X(v,.).$ Then X is strongly η-convex stochastic processes with modulus 1.

Theorem 6.

A random variable $X:I\times \Omega \to \mathbb{R}$ is an strongly η-convex stochastic process with modulus $\mu (.)>0$ if and only if for any ${\kappa }_1,{\kappa }_2,{\kappa }_3\in I$ with ${\kappa }_1\le {\kappa }_2\le {\kappa }_3,$ we have $|\begin{array}{ccc}0& 1& 1\\ ({\kappa }_3-{\kappa }_2)& X({\kappa }_2,.)-X({\kappa }_3,.)& 0\\ ({\kappa }_3-{\kappa }_1)& \eta (X({\kappa }_1,.),X({\kappa }_3,.))& \mu (.)({\kappa }_2-{\kappa }_1)({\kappa }_3-{\kappa }_1)\end{array}|\ge 0.$

Proof.

Suppose that X is an strongly η-convex stochastic process and ${\kappa }_1,{\kappa }_2,{\kappa }_3\in I$ with ${\kappa }_1\le {\kappa }_2\le {\kappa }_3.$ Then there exist $\lambda \in (0,1),$ such that ${\kappa }_2=\lambda {\kappa }_1+(1-\lambda ){\kappa }_3.$ $$\begin{array}{cc}X({\kappa }_2,.)=X(\lambda {\kappa }_1+(1-\lambda ){\kappa }_3,.)\hfill & \le X({\kappa }_3,.)+\left(\frac{{\kappa }_2-{\kappa }_3}{{\kappa }_1-{\kappa }_3}\right)\eta (X({\kappa }_1,.),X({\kappa }_3,.))\hfill \\ \hspace{0.17em}\hfill & -\mu (.)\left(\frac{{\kappa }_2-{\kappa }_3}{{\kappa }_1-{\kappa }_3}\right)\left(\frac{{\kappa }_1-{\kappa }_2}{{\kappa }_1-{\kappa }_3}\right){({\kappa }_3-{\kappa }_1)}^2.\hfill \end{array}$$

This implies, $$\begin{array}{cc}\hfill & (X({\kappa }_3,.)-X({\kappa }_2,.))({\kappa }_3-{\kappa }_1)+({\kappa }_3-{\kappa }_2)\eta (X({\kappa }_1,.),X({\kappa }_3,.))\hfill \\ \hspace{0.17em}\hfill & -\mu (.)({\kappa }_3-{\kappa }_2)({\kappa }_2-{\kappa }_1)({\kappa }_3-{\kappa }_1)\ge 0,\hfill \end{array}$$ or $|\begin{array}{ccc}0& 1& 1\\ ({\kappa }_3-{\kappa }_2)& X({\kappa }_2,.)-X({\kappa }_3,.)& 0\\ ({\kappa }_3-{\kappa }_1)& \eta (X({\kappa }_1,.),X({\kappa }_3,.))& \mu (.)({\kappa }_2-{\kappa }_1)({\kappa }_3-{\kappa }_1)\end{array}|\ge 0.$

For the converse part, take $u_1,u_2\in I$ with $u_1\le u_2.$ Choose any $\lambda \in (0,1),$ then we have $u_1\le \lambda u_1+(1-\lambda )u_2\le u_2.$

The above determinant is $$|\begin{array}{c}\begin{array}{ccc}0& 1& 1\\ (u_2-(\lambda u_1+(1-\lambda )u_2))& X(\lambda u_1+(1-\lambda )u_2,.)-X(u_2,.)& 0\\ (u_2-u_1)& \eta (X(u_1,.),X(u_2,.))& \mu (.)(\lambda u_1+(1-\lambda )u_2-u_1)(u_2-u_1)\end{array}|\\ \ge 0.\end{array}$$

This implies $$\begin{array}{c}X(\lambda u_1+(1-\lambda )u_2,.)\le X(u_2,.)+\lambda \eta (X(u_1,.),X(u_2,.))-\mu (.)\lambda (1-\lambda ){(u_2-u_1)}^2.\hfill \end{array}$$ □

Theorem 7.

Suppose that $X:[c,d]\times \Omega \to \mathbb{R}$ is an strongly η-convex stochastic process with modulus $\mu (.)>0$, such that η is bounded above on $X[c,d]\times X[c,d]$, then the following inequalities hold almost everywhere: $$\begin{array}{cc}\hfill & X\left(\frac{c+d}{2},.\right)-\frac{M_{\eta }}{2}+\frac{\mu (.)}{12}{(d-c)}^2\le \frac{1}{d-c}{\int }_c^{d}X(u,.)\mathit{\text{du}}\hfill \\ \hspace{0.17em}\hfill & \le \frac{X(c,.)+X(d,.)}{2}+\frac{1}{4}(\eta (X(c,.),X(d,.))+\eta (X(d,.),X(c,.)))-\frac{\mu (.)}{6}{(d-c)}^2\hfill \\ \hspace{0.17em}\hfill & \le \frac{X(c,.)+X(d,.)}{2}+M_{\eta }-\frac{\mu (.)}{6}{(d-c)}^2,\hfill \end{array}$$where $M_{\eta }$ is an upper bound of η.

Proof.

Since X is strongly η-convex stochastic process, therefore $$\begin{array}{cc}\hfill & X\left(\frac{c+d}{2},.\right)=X\left(\frac{1}{2}\left(\frac{c+d-\lambda (d-c)}{2}\right)+\frac{1}{2}\left(\frac{c+d+\lambda (d-c)}{2}\right),.\right)\hfill \\ \hspace{0.17em}\hfill & \le X\left(\frac{c+d+\lambda (d-c)}{2},.\right)+\frac{1}{2}\eta \left(X\left(\frac{c+d-\lambda (d-c)}{2},.\right),X\left(\frac{c+d+\lambda (d-c)}{2},.\right)\right)\hfill \\ \hspace{0.17em}\hfill & -\frac{\mu (.)}{4}{\lambda }^2{(d-c)}^2\le X\left(\frac{c+d+\lambda (d-c)}{2},.\right)+\frac{M_{\eta }}{2}-\frac{\mu (.)}{4}{\lambda }^2{(d-c)}^2.\hfill \end{array}$$

This implies, $$\begin{array}{c}X\left(\frac{c+d}{2},.\right)-\frac{M_{\eta }}{2}+\frac{\mu (.)}{4}{\lambda }^2{(d-c)}^2\le X\left(\frac{c+d+\lambda (d-c)}{2},.\right).\hfill \end{array}$$

Similarly, $$\begin{array}{c}X\left(\frac{c+d}{2},.\right)-\frac{M_{\eta }}{2}+\frac{\mu (.)}{4}{\lambda }^2{(d-c)}^2\le X\left(\frac{c+d-\lambda (d-c)}{2},.\right).\hfill \end{array}$$

By using the change of variable technique, we have (1) $$\begin{array}{l}\frac{1}{d-c}{\int }_c^{d}X(u,.)\mathit{\text{du}}=\frac{1}{d-c}\left[{\int }_c^{(c+d)/2}X(u,.)\mathit{\text{du}}+{\int }_{(c+d)/2}^{d}X(u,.)\mathit{\text{du}}\right]\\ =\frac{1}{2}{\int }_0^{1}X\left(\frac{c+d-\lambda (d-c)}{2},.\right)d\lambda +\frac{1}{2}{\int }_0^{1}X\left(\frac{c+d+\lambda (d-c)}{2},.\right)d\lambda \\ \ge {\int }_0^1\left[X\left(\frac{c+d}{2},.\right)-\frac{M_{\eta }}{2}+\frac{\mu (.)}{4}{\lambda }^2{(d-c)}^2\right]d\lambda \\ =X\left(\frac{c+d}{2},.\right)-\frac{M_{\eta }}{2}+\frac{\mu (.)}{12}{(d-c)}^2.\end{array}$$(1)

We now prove the right hand side of the theorem. Since X is strongly η-convex stochastic process with modulus $\mu (.)>0,$ we get $$\begin{array}{c}X(\lambda c+(1-\lambda )d,.)\le X(d,.)+\lambda \eta (X(c,.),X(d,.))-\mu (.)\lambda (1-\lambda ){(d-c)}^2.\hfill \end{array}$$

Integrating above inequality with respect to λ on both sides from 0 to 1, we have $$\begin{array}{c}{\int }_0^{1}X(\lambda c+(1-\lambda )d,.)d\lambda \le {\int }_0^1(X(d,.)+\lambda \eta (X(c,.),X(d,.))-\mu (.)\lambda (1-\lambda ){(d-c)}^2)d\lambda .\hfill \end{array}$$

This implies, $$\begin{array}{c}\frac{1}{d-c}{\int }_c^{d}X(u,.)\mathit{\text{du}}\le X(d,.)+\frac{1}{2}\eta (X(c,.),X(d,.))-\frac{\mu (.)}{6}{(d-c)}^2=P.\hfill \end{array}$$

Similarly, $$\begin{array}{c}\frac{1}{d-c}{\int }_c^{d}X(u,.)\mathit{\text{du}}\le X(c,.)+\frac{1}{2}\eta (X(d,.),X(c,.))-\frac{\mu (.)}{6}{(d-c)}^2=Q.\hfill \end{array}$$

Therefore, (2) $$\begin{array}{l}\frac{1}{d-c}{\int }_c^{d}X(u,.)\mathit{\text{du}}\le \mathrm{\text{Min}}\{P,Q\}\\ \le \frac{X(c,.)+X(d,.)}{2}+\frac{1}{4}(\eta (X(c,.),X(d,.))+\eta (X(d,.),X(c,.)))\\ -\frac{\mu (.)}{6}{(d-c)}^2\\ \le \frac{X(c,.)+X(d,.)}{2}+M_{\eta }-\frac{\mu (.)}{6}{(d-c)}^2.\end{array}$$(2)

From Equations (1)(1) $$\begin{array}{l}\frac{1}{d-c}{\int }_c^{d}X(u,.)\mathit{\text{du}}=\frac{1}{d-c}\left[{\int }_c^{(c+d)/2}X(u,.)\mathit{\text{du}}+{\int }_{(c+d)/2}^{d}X(u,.)\mathit{\text{du}}\right]\\ =\frac{1}{2}{\int }_0^{1}X\left(\frac{c+d-\lambda (d-c)}{2},.\right)d\lambda +\frac{1}{2}{\int }_0^{1}X\left(\frac{c+d+\lambda (d-c)}{2},.\right)d\lambda \\ \ge {\int }_0^1\left[X\left(\frac{c+d}{2},.\right)-\frac{M_{\eta }}{2}+\frac{\mu (.)}{4}{\lambda }^2{(d-c)}^2\right]d\lambda \\ =X\left(\frac{c+d}{2},.\right)-\frac{M_{\eta }}{2}+\frac{\mu (.)}{12}{(d-c)}^2.\end{array}$$(1) and (2)(2) $$\begin{array}{l}\frac{1}{d-c}{\int }_c^{d}X(u,.)\mathit{\text{du}}\le \mathrm{\text{Min}}\{P,Q\}\\ \le \frac{X(c,.)+X(d,.)}{2}+\frac{1}{4}(\eta (X(c,.),X(d,.))+\eta (X(d,.),X(c,.)))\\ -\frac{\mu (.)}{6}{(d-c)}^2\\ \le \frac{X(c,.)+X(d,.)}{2}+M_{\eta }-\frac{\mu (.)}{6}{(d-c)}^2.\end{array}$$(2) , we have (3) $$\begin{array}{l}X\left(\frac{c+d}{2},.\right)-\frac{M_{\eta }}{2}+\frac{\mu (.)}{12}{(d-c)}^2\le \frac{1}{d-c}{\int }_c^{d}X(u,.)\mathit{\text{du}}\\ \le \frac{X(c,.)+X(d,.)}{2}+\frac{1}{4}(\eta (X(c,.),X(d,.))+\eta (X(d,.),X(c,.)))-\frac{\mu (.)}{6}{(d-c)}^2\\ \le \frac{X(c,.)+X(d,.)}{2}+M_{\eta }-\frac{\mu (.)}{6}{(d-c)}^2.\end{array}$$(3) □

Remark 2.

When $\mu (.)=0$, then above theorem reduces to Theorem 5. If we consider $\eta (X(u,.),X(v,.))=X(u,.)-X(v,.)\hspace{0.17em}\text{and}\hspace{0.17em}\mu (.)=0$, then above theorem reduces to the classical Hermite-Hadamard inequality for convex stochastic process (Kotrys 2012).

Theorem 8.

If a stochastic process $X:I\times \Omega \to \mathbb{R}$ be an strongly η-convex with modulus $\mu (.)>0$ and integrable on $I\times \Omega ,$ we have $$\begin{array}{ll}\hfill & \frac{1}{d-c}{\int }_c^{d}X(u,.)X(c+d-u,.)\mathit{\text{du}}\le X(c,.)X(d,.)+\frac{1}{2}(X(c,.)\eta (X(c,.),X(d,.))\hfill \\ \hspace{0.17em}\hfill & +X(d,.)\eta (X(d,.),X(c,.)))+\frac{1}{3}\eta (X(c,.),X(d,.))\eta (X(d,.),X(c,.))\hfill \\ \hspace{0.17em}\hfill & -\frac{\mu (.)}{12}{(d-c)}^2(\eta (X(c,.),X(d,.))+\eta (X(d,.),X(c,.)))\hfill \\ \hspace{0.17em}\hfill & -\frac{\mu (.)}{6}{(d-c)}^2(X(c,.)+X(d,.))+\frac{{\mu }^2(.)}{30}{(d-c)}^4.\hfill \end{array}$$

Proof.

Since X is strongly η-convex stochastic process, therefore (4) $$X(\lambda c+(1-\lambda )d,.)\le X(d,.)+\lambda \eta (X(c,.),X(d,.))-\mu (.)\lambda (1-\lambda ){(d-c)}^2$$(4) and (5) $$X(\lambda d+(1-\lambda )c,.)\le X(c,.)+\lambda \eta (X(d,.),X(c,.))-\mu (.)\lambda (1-\lambda ){(d-c)}^2.$$(5)

From Equations (4)(4) $$X(\lambda c+(1-\lambda )d,.)\le X(d,.)+\lambda \eta (X(c,.),X(d,.))-\mu (.)\lambda (1-\lambda ){(d-c)}^2$$(4) and (5)(5) $$X(\lambda d+(1-\lambda )c,.)\le X(c,.)+\lambda \eta (X(d,.),X(c,.))-\mu (.)\lambda (1-\lambda ){(d-c)}^2.$$(5) , we obtain $$\begin{array}{ll}\hfill & X(\lambda c+(1-\lambda )d,.)X(\lambda d+(1-\lambda )c,.)\le X(c,.)X(d,.)+\lambda (X(c,.)\eta (X(c,.),X(d,.))\hfill \\ \hspace{0.17em}\hfill & +X(d,.)\eta (X(d,.),X(c,.)))+{\lambda }^2\eta (X(c,.),X(d,.))\eta (X(d,.),X(c,.))\hfill \\ \hspace{0.17em}\hfill & -\mu (.)\lambda (1-\lambda ){(d-c)}^2(X(c,.)+X(d,.))-\mu (.){\lambda }^2(1-\lambda ){(d-c)}^2(\eta (X(c,.),X(d,.))\hfill \\ \hspace{0.17em}\hfill & +\eta (X(d,.),X(c,.)))+{\mu }^2(.){\lambda }^2{(1-\lambda )}^2{(d-c)}^4.\hfill \end{array}$$

Integrating above inequality from 0 to 1 on both sides with respect to $\lambda ,$ we have $$\begin{array}{cc}\hfill & {\int }_0^{1}X(\lambda c+(1-\lambda )d,.)X(\lambda d+(1-\lambda )c,.)d\lambda \le X(c,.)X(d,.)+\frac{1}{2}(X(c,.)\eta (X(c,.),X(d,.))\hfill \\ \hspace{0.17em}\hfill & +X(d,.)\eta (X(d,.),X(c,.)))+\frac{1}{3}\eta (X(c,.),X(d,.))\eta (X(d,.),X(c,.))\hfill \\ \hspace{0.17em}\hfill & -\frac{\mu (.)}{12}{(d-c)}^2(\eta (X(c,.),X(d,.))+\eta (X(d,.),X(c,.)))\hfill \\ \hspace{0.17em}\hfill & -\frac{\mu (.)}{6}{(d-c)}^2(X(c,.)+X(d,.))+\frac{{\mu }^2(.)}{30}{(d-c)}^4.\hfill \end{array}$$

This implies, $$\begin{array}{cc}\hfill & \frac{1}{d-c}{\int }_c^{d}X(u,.)X(c+d-u,.)\mathit{\text{du}}\le X(c,.)X(d,.)+\frac{1}{2}(X(c,.)\eta (X(c,.),X(d,.))\hfill \\ \hspace{0.17em}\hfill & +X(d,.)\eta (X(d,.),X(c,.)))+\frac{1}{3}\eta (X(c,.),X(d,.))\eta (X(d,.),X(c,.))\hfill \\ \hspace{0.17em}\hfill & -\frac{\mu (.)}{12}{(d-c)}^2(\eta (X(c,.),X(d,.))+\eta (X(d,.),X(c,.)))\hfill \\ \hspace{0.17em}\hfill & -\frac{\mu (.)}{6}{(d-c)}^2(X(c,.)+X(d,.))+\frac{{\mu }^2(.)}{30}{(d-c)}^4.\hfill \end{array}$$ □

Remark 3.

When X is strongly log-convex stochastic process, then above theorem reduces to Theorem 3 of (Tomar, Set, and Bekar 2014).

Theorem 9.

Let $X:I\times \Omega \to \mathbb{R}$ be a mean square stochastic process such that $X\prime$ is mean square integrable on $[c,d],$ where $c,d\in I$ with $c<d.$ If $|X\prime |$ is an strongly η-convex stochastic process with modulus $\mu (.)>0$ on I and $|X\prime (t,.)|\le M$ for every t, then $$\begin{array}{cc}\hfill & |X(t,.)-\frac{1}{d-c}{\int }_c^{d}X(u,.)\mathit{\text{du}}|\hfill \\ \hspace{0.17em}\hfill & \le \frac{M}{2(d-c)}({(t-c)}^2+{(d-t)}^2)+\frac{1}{3(d-c)}[{(t-c)}^2\eta (|X\prime (t,.)|,|X\prime (c,.)|)\hfill \\ \hspace{0.17em}\hfill & +{(d-t)}^2\eta (|X\prime (t,.)|,|X\prime (d,.)|)]-\frac{\mu (.)}{12(d-c)}({(t-c)}^4+{(d-t)}^4).\hfill \end{array}$$

Proof.

Recall Lemma 1: $$\begin{array}{cc}X(t,.)-\frac{1}{d-c}{\int }_c^{d}X(u,.)\mathit{\text{du}}\hfill & =\frac{{(t-c)}^2}{d-c}{\int }_0^{1}yX\prime (\mathit{\text{yt}}+(1-y)c,.)\mathit{\text{dy}}\hfill \\ \hspace{0.17em}\hfill & -\frac{{(d-t)}^2}{d-c}{\int }_0^{1}yX\prime (\mathit{\text{yt}}+(1-y)d,.)\mathit{\text{dy}},\hspace{0.17em}(a.e),\hspace{0.17em}\text{for}\mathrm{ }\text{each}\hspace{0.17em}t\in [c,d].\hfill \end{array}$$

Since $|X\prime |$ is strongly η-convex stochastic process, therefore $$\begin{array}{cc}\hfill & \left|X(t,.)-\frac{1}{d-c}{\int }_c^{d}X(u,.)\mathit{\text{du}}\right|\hfill \\ \hspace{0.17em}\hfill & \le \frac{{(t-c)}^2}{d-c}{\int }_0^{1}y\left[\right|X\prime (c,.)|+y\eta (|X\prime (t,.)|,|X\prime (c,.)|)-\mu (.)y(1-y){(c-t)}^2]\mathit{\text{dy}}\hfill \\ \hspace{0.17em}\hfill & +\frac{{(d-t)}^2}{d-c}{\int }_0^{1}y\left[\right|X\prime (d,.)|+y\eta (|X\prime (t,.)|,|X\prime (d,.)|)-\mu (.)y(1-y){(d-t)}^2]\mathit{\text{dy}}\hfill \\ \hspace{0.17em}\hfill & \le \frac{{(t-c)}^2}{d-c}\left[\frac{M}{2}+\frac{\eta (|X\prime (t,.)|,|X\prime (c,.)|)}{3}-\frac{\mu (.)}{12}{(c-t)}^2\right]\hfill \\ \hspace{0.17em}\hfill & +\frac{{(d-t)}^2}{d-c}\left[\frac{M}{2}+\frac{\eta (|X\prime (t,.)|,|X\prime (d,.)|)}{3}-\frac{\mu (.)}{12}{(d-t)}^2\right]\hfill \\ \hspace{0.17em}\hfill & =\frac{M}{2(d-c)}({(t-c)}^2+{(d-t)}^2)+\frac{1}{3(d-c)}[{(t-c)}^2\eta (|X\prime (t,.)|,|X\prime (c,.)|)\hfill \\ \hspace{0.17em}\hfill & +{(d-t)}^2\eta (|X\prime (t,.)|,|X\prime (d,.)|)]-\frac{\mu (.)}{12(d-c)}({(t-c)}^4+{(d-t)}^4).\hfill \end{array}$$ □

Remark 4.

When $\mu (.)=0$, then above theorem reduces to Theorem 4.2 of (Jung et al. 2021).

Theorem 10.

Let $X:I\times \Omega \to \mathbb{R}$ be a mean square differentiable stochastic process on I⁰ and $X\prime$ be a mean square integrable on $[c,d]$, where $c,d\in I,c<d.$ If $|X\prime |$ is an strongly η-convex stochastic process on $[c,d],$ then we have almost everywhere: $$\begin{array}{cc}\left|\frac{X(c,.)+X(d,.)}{2}-\frac{1}{d-c}{\int }_c^{d}X(u,.)\mathit{\text{du}}\right|\hfill & \le \frac{d-c}{4}(|X\prime (d,.)|+\frac{1}{2}\eta (|X\prime (c,.)|,|X\prime (d,.)|)\hfill \\ \hspace{0.17em}\hfill & -\frac{\mu (.)}{8}{(d-c)}^2).\hfill \end{array}$$

Proof.

From Lemma 2, we have $$\begin{array}{c}\left|\frac{X(c,.)+X(d,.)}{2}-\frac{1}{d-c}{\int }_c^{d}X(u,.)\mathit{\text{du}}\right|\le \frac{d-c}{2}{\int }_0^1|1-2\lambda \left|\right|X\prime (\lambda c+(1-\lambda )d,.)|d\lambda .\hfill \end{array}$$

Using the definition of strong η-convex stochastic process in above inequality, we have $$\begin{array}{cc}\hfill & \left|\frac{X(c,.)+X(d,.)}{2}-\frac{1}{d-c}{\int }_c^{d}X(u,.)\mathit{\text{du}}\right|\hfill \\ \hspace{0.17em}\hfill & \le \frac{d-c}{2}{\int }_0^1|1-2\lambda |(|X\prime (d,.)|+\lambda \eta (|X\prime (c,.)|,|X\prime (d,.)|)-\mu (.)\lambda (1-\lambda ){(d-c)}^2)d\lambda \hfill \\ \hspace{0.17em}\hfill & =\frac{d-c}{2}({\int }_0^1|1-2\lambda \left|\right|X\prime (d,.)|d\lambda +{\int }_0^1\lambda |1-2\lambda |\eta (|X\prime (c,.)|,|X\prime (d,.)|)d\lambda \hfill \\ \hspace{0.17em}\hfill & -\mu (.){(d-c)}^2{\int }_0^1\lambda (1-\lambda )|1-2\lambda |d\lambda )\hfill \\ \hspace{0.17em}\hfill & =\frac{d-c}{2}(\frac{1}{2}|X\prime (d,.)|+\frac{1}{4}\eta (|X\prime (c,.)|,|X\prime (d,.)|)-\frac{\mu (.)}{16}{(d-c)}^2)\hfill \\ \hspace{0.17em}\hfill & =\frac{d-c}{4}(|X\prime (d,.)|+\frac{1}{2}\eta (|X\prime (c,.)|,|X\prime (d,.)|)-\frac{\mu (.)}{8}{(d-c)}^2).\hfill \end{array}$$ □

Corollary 1.

When $\mu (.)=0$ in above theorem, then we obtain the following inequality for η-convex stochastic processes: $$\begin{array}{cc}\left|\frac{X(c,.)+X(d,.)}{2}-\frac{1}{d-c}{\int }_c^{d}X(u,.)\mathit{\text{du}}\right|\hfill & \le \frac{d-c}{4}\left(|X\prime (d,.)|+\frac{1}{2}\eta (|X\prime (c,.)|,|X\prime (d,.)|)\right)\hspace{0.17em}\hspace{0.17em}(a.e.).\hfill \end{array}$$

Remark 5.

When $\mu (.)=0$ and $\eta (|X\prime (c,.)|,|X\prime (d,.)|)=|X\prime (c,.)|-|X\prime (d,.)|,$ then above theorem reduces to Corollary 5.3 of (Fu et al. 2021).

Theorem 11.

Let $X:I\times \Omega \to \mathbb{R}$ be a mean square differentiable stochastic process on I⁰ with $q>1,\hspace{0.17em}\frac{1}{p}+\frac{1}{q}=1$ and assume that $X\prime$ be a mean square integrable on $[c,d]$, where $c,d\in I,c<d.$ If $|X\prime {|}^q$ is an strongly η-convex stochastic process on $[c,d],$ then we have almost everywhere: $$\begin{array}{cc}\hfill & \left|\frac{X(c,.)+X(d,.)}{2}-\frac{1}{d-c}{\int }_c^{d}X(u,.)\mathit{\text{du}}\right|\hfill \\ \hspace{0.17em}\hfill & \le \frac{d-c}{2}{\left(\frac{1}{p+1}\right)}^{1/p}{\left(|X\prime (d,.){|}^q+\frac{1}{2}\eta (|X\prime (c,.){|}^q,|X\prime (d,.){|}^q)-\frac{\mu (.)}{6}{(d-c)}^2\right)}^{1/q}.\hfill \end{array}$$

Proof.

From Lemma 2, we have $$\begin{array}{c}\left|\frac{X(c,.)+X(d,.)}{2}-\frac{1}{d-c}{\int }_c^{d}X(u,.)\mathit{\text{du}}\right|\le \frac{d-c}{2}{\int }_0^1|1-2\lambda \left|\right|X\prime (\lambda c+(1-\lambda )d,.)|d\lambda .\hfill \end{array}$$

Using Hölder’s inequality and the definition of strong η-convex stochastic process in above inequality, we obtain $$\begin{array}{cc}\hfill & \left|\frac{X(c,.)+X(d,.)}{2}-\frac{1}{d-c}{\int }_c^{d}X(u,.)\mathit{\text{du}}\right|\hfill \\ \hspace{0.17em}\hfill & \le \frac{d-c}{2}{\left({\int }_0^1|1-2\lambda {|}^{p}d\lambda \right)}^{1/p}{\left({\int }_0^1|X\prime (\lambda c+(1-\lambda )d,.){|}^{q}d\lambda \right)}^{1/q}\hfill \\ \hspace{0.17em}\hfill & \le \frac{d-c}{2}{\left(\frac{1}{p+1}\right)}^{1/p}{\left({\int }_0^1(|X\prime (d,.){|}^q+\lambda \eta (|X\prime (c,.){|}^q,|X\prime (d,.){|}^q)-\mu (.)\lambda (1-\lambda ){(d-c)}^2)d\lambda \right)}^{1/q}\hfill \\ \hspace{0.17em}\hfill & =\frac{d-c}{2}{\left(\frac{1}{p+1}\right)}^{1/p}{\left(|X\prime (d,.){|}^q+\frac{1}{2}\eta (|X\prime (c,.){|}^q,|X\prime (d,.){|}^q)-\frac{\mu (.)}{6}{(d-c)}^2\right)}^{1/q}.\hfill \end{array}$$ □

Corollary 2.

When $\mu (.)=0$ in above theorem, then we obtain the following inequality for η-convex stochastic processes: $$\begin{array}{cc}\hfill & \left|\frac{X(c,.)+X(d,.)}{2}-\frac{1}{d-c}{\int }_c^{d}X(u,.)\mathit{\text{du}}\right|\hfill \\ \hspace{0.17em}\hfill & \le \frac{d-c}{2}{\left(\frac{1}{p+1}\right)}^{1/p}{\left(|X\prime (d,.){|}^q+\frac{1}{2}\eta (|X\prime (c,.){|}^q,|X\prime (d,.){|}^q)\right)}^{1/q}\hspace{0.17em}\hspace{0.17em}(a.e.).\hfill \end{array}$$

Remark 6.

When $\mu (.)=0$ and $\eta (|X\prime (c,.){|}^q,|X\prime (d,.){|}^q)=|X\prime (c,.){|}^q-|X\prime (d,.){|}^q,$ then above theorem reduces to Corollary 5.5 of (Fu et al. 2021).

Theorem 12.

Let $X:I\times \Omega \to \mathbb{R}$ be a mean square differentiable stochastic process on I⁰ with $q\ge 1,$ and assume that $X\prime$ be a mean square integrable on $[c,d]$, where $c,d\in I,c<d.$ If $|X\prime {|}^q$ is an strongly η-convex stochastic process on $[c,d],$ then we have almost everywhere: $$\begin{array}{cc}\hfill & \left|\frac{X(c,.)+X(d,.)}{2}-\frac{1}{d-c}{\int }_c^{d}X(u,.)\mathit{\text{du}}\right|\hfill \\ \hspace{0.17em}\hfill & \le \frac{d-c}{4}{\left(|X\prime (d,.){|}^q+\frac{1}{2}\eta (|X\prime (c,.){|}^q,|X\prime (d,.){|}^q)-\frac{\mu (.)}{8}{(d-c)}^2\right)}^{1/q}.\hfill \end{array}$$

Proof.

For $q=1,$ we use the estimates from the proof of Theorem 10. Now we prove result for $q>1.$ From Lemma 2, we have $$\begin{array}{c}\left|\frac{X(c,.)+X(d,.)}{2}-\frac{1}{d-c}{\int }_c^{d}X(u,.)\mathit{\text{du}}\right|\le \frac{d-c}{2}{\int }_0^1|1-2\lambda \left|\right|X\prime (\lambda c+(1-\lambda )d,.)|d\lambda .\hfill \end{array}$$

Using Hölder’s inequality and the definition of strong η-convex stochastic process in above inequality, we obtain $$\begin{array}{cc}\hfill & \left|\frac{X(c,.)+X(d,.)}{2}-\frac{1}{d-c}{\int }_c^{d}X(u,.)\mathit{\text{du}}\right|\hfill \\ \hspace{0.17em}\hfill & \le \frac{d-c}{2}{\left({\int }_0^1|1-2\lambda |d\lambda \right)}^{1-\frac{1}{q}}{\left({\int }_0^1|1-2\lambda \left|\right|X\prime (\lambda c+(1-\lambda )d,.){|}^{q}d\lambda \right)}^{1/q}\hfill \\ \hspace{0.17em}\hfill & \le \frac{d-c}{2}{\left(\frac{1}{2}\right)}^{1-\frac{1}{q}}({\int }_0^1|1-2\lambda |(|X\prime (d,.){|}^q+\lambda \eta (|X\prime (c,.){|}^q,|X\prime (d,.){|}^q)\hfill \\ \hspace{0.17em}\hfill & -{\mu (.)\lambda (1-\lambda ){(d-c)}^2)d\lambda )}^{1/q}\hfill \\ \hspace{0.17em}\hfill & =\frac{d-c}{2}{(\frac{1}{2})}^{1-\frac{1}{q}}{(\frac{1}{2}|X\prime (d,.){|}^q+\frac{1}{4}\eta (|X\prime (c,.){|}^q,|X\prime (d,.){|}^q)-\frac{\mu (.)}{16}{(d-c)}^2)}^{1/q}\hfill \\ \hspace{0.17em}\hfill & =\frac{d-c}{4}{(|X\prime (d,.){|}^q+\frac{1}{2}\eta (|X\prime (c,.){|}^q,|X\prime (d,.){|}^q)-\frac{\mu (.)}{8}{(d-c)}^2)}^{1/q}.\hfill \end{array}$$ □

Corollary 3.

When $\mu (.)=0$ in above theorem, then we obtain the following inequality for η-convex stochastic processes: $$\begin{array}{cc}\hfill & \left|\frac{X(c,.)+X(d,.)}{2}-\frac{1}{d-c}{\int }_c^{d}X(u,.)\mathit{\text{du}}\right|\hfill \\ \hspace{0.17em}\hfill & \le \frac{d-c}{4}{\left(|X\prime (d,.){|}^q+\frac{1}{2}\eta (|X\prime (c,.){|}^q,|X\prime (d,.){|}^q)\right)}^{1/q}\hspace{0.17em}\hspace{0.17em}(a.e.).\hfill \end{array}$$

Remark 7.

When $\mu (.)=0$ and $\eta (|X\prime (c,.){|}^q,|X\prime (d,.){|}^q)=|X\prime (c,.){|}^q-|X\prime (d,.){|}^q,$ then above theorem reduces to Corollary 5.8 of (Fu et al. 2021).

Theorem 13.

Let $X:I\times \Omega \to \mathbb{R}$ be a mean square differentiable stochastic process on I⁰ with $q>1,\hspace{0.17em}\frac{1}{p}+\frac{1}{q}=1$ and assume that $X\prime$ be a mean square integrable on $[c,d]$, where $c,d\in I,c<d.$ If $|X\prime {|}^q$ is an strongly η-convex stochastic process on $[c,d],$ then we have almost everywhere: $$\begin{array}{cc}\hfill & \left|\frac{X(c,.)+X(d,.)}{2}-\frac{1}{d-c}{\int }_c^{d}X(u,.)\mathit{\text{du}}\right|\hfill \\ \hspace{0.17em}\hfill & \le \frac{d-c}{2}{\left(\frac{1}{2(p+1)}\right)}^{\frac{1}{p}}[{\left(\frac{1}{2}|X\prime (d,.){|}^q+\frac{1}{6}\eta (|X\prime (c,.){|}^q,|X\prime (d,.){|}^q)-\frac{\mu (.)}{12}{(d-c)}^2\right)}^{1/q}\hfill \\ \hspace{0.17em}\hfill & +{\left(\frac{1}{2}(|X\prime (d,.){|}^q+\frac{1}{3}\eta (|X\prime (c,.){|}^q,|X\prime (d,.){|}^q)-\frac{\mu (.)}{12}{(d-c)}^2\right)}^{1/q}].\hfill \end{array}$$

Proof.

From Lemma 2, we have $$\begin{array}{c}\left|\frac{X(c,.)+X(d,.)}{2}-\frac{1}{d-c}{\int }_c^{d}X(u,.)\mathit{\text{du}}\right|\le \frac{d-c}{2}{\int }_0^1|1-2\lambda \left|\right|X\prime (\lambda c+(1-\lambda )d,.)|d\lambda .\hfill \end{array}$$

Using Hölder’s inequality and the definition of strong η-convex stochastic process in above inequality, we get $$\begin{array}{cc}\hfill & \left|\frac{X(c,.)+X(d,.)}{2}-\frac{1}{d-c}{\int }_c^{d}X(u,.)\mathit{\text{du}}\right|\hfill \\ \hspace{0.17em}\hfill & \le \frac{d-c}{2}{({\int }_0^1(1-\lambda )|1-2\lambda {|}^{p}d\lambda )}^{\frac{1}{p}}{({\int }_0^1(1-\lambda )|X\prime (\lambda c+(1-\lambda )d,.){|}^{q}d\lambda )}^{1/q}\hfill \\ \hspace{0.17em}\hfill & +\frac{d-c}{2}{({\int }_0^1\lambda |1-2\lambda {|}^{p}d\lambda )}^{\frac{1}{p}}{({\int }_0^1\lambda |X\prime (\lambda c+(1-\lambda )d,.){|}^{q}d\lambda )}^{1/q}\hfill \\ \hspace{0.17em}\hfill & \le \frac{d-c}{2}{(\frac{1}{2(p+1)})}^{\frac{1}{p}}\hfill \\ \hspace{0.17em}\hfill & \times [{({\int }_0^1(1-\lambda )(|X\prime (d,.){|}^q+\lambda \eta (|X\prime (c,.){|}^q,|X\prime (d,.){|}^q)-\mu (.)\lambda (1-\lambda ){(d-c)}^2)d\lambda )}^{1/q}\hfill \\ \hspace{0.17em}\hfill & +{({\int }_0^1\lambda (|X\prime (d,.){|}^q+\lambda \eta (|X\prime (c,.){|}^q,|X\prime (d,.){|}^q)-\mu (.)\lambda (1-\lambda ){(d-c)}^2)d\lambda )}^{1/q}]\hfill \\ \hspace{0.17em}\hfill & =\frac{d-c}{2}{(\frac{1}{2(p+1)})}^{\frac{1}{p}}[{(\frac{1}{2}|X\prime (d,.){|}^q+\frac{1}{6}\eta (|X\prime (c,.){|}^q,|X\prime (d,.){|}^q)-\frac{\mu (.)}{12}{(d-c)}^2)}^{1/q}\hfill \\ \hspace{0.17em}\hfill & +{(\frac{1}{2}(|X\prime (d,.){|}^q+\frac{1}{3}\eta (|X\prime (c,.){|}^q,|X\prime (d,.){|}^q)-\frac{\mu (.)}{12}{(d-c)}^2)}^{1/q}].\hfill \end{array}$$ □

Corollary 4.

When $\mu (.)=0$ in above theorem, then we obtain the following inequality for η-convex stochastic processes: $$\begin{array}{cc}\hfill & \left|\frac{X(c,.)+X(d,.)}{2}-\frac{1}{d-c}{\int }_c^{d}X(u,.)\mathit{\text{du}}\right|\hfill \\ \hspace{0.17em}\hfill & \le \frac{d-c}{2}{(\frac{1}{2(p+1)})}^{\frac{1}{p}}[{(\frac{1}{2}|X\prime (d,.){|}^q+\frac{1}{6}\eta (|X\prime (c,.){|}^q,|X\prime (d,.){|}^q))}^{1/q}\hfill \\ \hspace{0.17em}\hfill & +{(\frac{1}{2}(|X\prime (d,.){|}^q+\frac{1}{3}\eta (|X\prime (c,.){|}^q,|X\prime (d,.){|}^q))}^{1/q}]\hspace{0.17em}\hspace{0.17em}(a.e.).\hfill \end{array}$$

Remark 8.

When $\mu (.)=0$ and $\eta (|X\prime (c,.){|}^q,|X\prime (d,.){|}^q)=|X\prime (c,.){|}^q-|X\prime (d,.){|}^q,$ then above theorem reduces to Corollary 5.10 of (Fu et al. 2021).

Theorem 14.

Let $X:I\times \Omega \to \mathbb{R}$ be a mean square differentiable stochastic process on I⁰ with $q\ge 1,$ and assume that $X\prime$ be a mean square integrable on $[c,d]$, where $c,d\in I,c<d.$ If $|X\prime {|}^q$ is an strongly η-convex stochastic process on $[c,d],$ then we have almost everywhere: $$\begin{array}{cc}\hfill & \left|\frac{X(c,.)+X(d,.)}{2}-\frac{1}{d-c}{\int }_c^{d}X(u,.)\mathit{\text{du}}\right|\le \frac{d-c}{8}[(|X\prime (d,.){|}^q+\frac{1}{4}\eta (|X\prime (c,.){|}^q,|X\prime (d,.){|}^q)\hfill \\ \hspace{0.17em}\hfill & {-\frac{\mu (.)}{8}{(d-c)}^2)}^{1/q}+{(|X\prime (d,.){|}^q+\frac{3}{4}\eta (|X\prime (c,.){|}^q,|X\prime (d,.){|}^q)-\frac{\mu (.)}{8}{(d-c)}^2)}^{1/q}].\hfill \end{array}$$

Proof.

For $q=1,$ we use the estimates from the proof of Theorem 10. Now we prove result for $q>1.$ From Lemma 2, we have $$\begin{array}{c}\left|\frac{X(c,.)+X(d,.)}{2}-\frac{1}{d-c}{\int }_c^{d}X(u,.)\mathit{\text{du}}\right|\le \frac{d-c}{2}{\int }_0^1|1-2\lambda \left|\right|X\prime (\lambda c+(1-\lambda )d,.)|d\lambda .\hfill \end{array}$$

Using improved power-mean integral inequality and the definition of strong η-convex stochastic process in above inequality, we get $$\begin{array}{cc}\hfill & \left|\frac{X(c,.)+X(d,.)}{2}-\frac{1}{d-c}{\int }_c^{d}X(u,.)\mathit{\text{du}}\right|\hfill \\ \hspace{0.17em}\hfill & \le \frac{d-c}{2}{({\int }_0^1(1-\lambda )|1-2\lambda |d\lambda )}^{1-\frac{1}{q}}{({\int }_0^1(1-\lambda )|1-2\lambda ||X^{\prime }(\lambda c+(1-\lambda )d,.){|}^{q}d\lambda )}^{1/q}\hfill \\ \hspace{0.17em}\hfill & +\frac{d-c}{2}{({\int }_0^1\lambda |1-2\lambda |d\lambda )}^{1-\frac{1}{q}}{({\int }_0^1\lambda |1-2\lambda ||X^{\prime }(\lambda c+(1-\lambda )d,.){|}^{q}d\lambda )}^{1/q}\hfill \\ \hfill & \le \frac{d-c}{2}{(\frac{1}{4})}^{1-\frac{1}{q}}[({\int }_0^1(1-\lambda )|1-2\lambda |(|X^{\prime }(d,.){|}^q+\lambda \eta (|X^{\prime }(c,.){|}^q,|X^{\prime }(d,.){|}^q)\hfill \\ \hspace{0.17em}\hfill & -{\mu (.)\lambda (1-\lambda ){(d-c)}^2)d\lambda )}^{1/q}+({\int }_0^1\lambda |1-2\lambda |(|X^{\prime }(d,.){|}^q+\lambda \eta (|X^{\prime }(c,.){|}^q,|X^{\prime }(d,.){|}^q)\hfill \\ \hspace{0.17em}\hfill & -{\mu (.)\lambda (1-\lambda ){(d-c)}^2)d\lambda )}^{1/q}]\hfill \\ \hspace{0.17em}\hfill & =\frac{d-c}{2}{(\frac{1}{4})}^{1-\frac{1}{q}}[{(\frac{1}{4}|X^{\prime }(d,.){|}^q+\frac{1}{16}\eta (|X^{\prime }(c,.){|}^q,|X^{\prime }(d,.){|}^q)-\frac{\mu (.)}{32}{(d-c)}^2)}^{1/q}\hfill \\ \hspace{0.17em}\hfill & +{(\frac{1}{4}|X^{\prime }(d,.){|}^q+\frac{3}{16}\eta (|X^{\prime }(c,.){|}^q,|X^{\prime }(d,.){|}^q)-\frac{\mu (.)}{32}{(d-c)}^2)}^{1/q}]\hfill \\ \mathrm{ }\hfill & =\frac{d-c}{8}[{(|X^{\prime }(d,.){|}^q+\frac{1}{4}\eta (|X^{\prime }(c,.){|}^q,|X^{\prime }(d,.){|}^q)-\frac{\mu (.)}{8}{(d-c)}^2)}^{1/q}\hfill \\ \hspace{0.17em}\hfill & +{(|X^{\prime }(d,.){|}^q+\frac{3}{4}\eta (|X^{\prime }(c,.){|}^q,|X^{\prime }(d,.){|}^q)-\frac{\mu (.)}{8}{(d-c)}^2)}^{1/q}].\hfill \end{array}$$ □

Corollary 5.

When $\mu (.)=0$ in above theorem, then we obtain the following inequality for η-convex stochastic processes: $$\begin{array}{cc}\hfill & \left|\frac{X(c,.)+X(d,.)}{2}-\frac{1}{d-c}{\int }_c^{d}X(u,.)\mathit{\text{du}}\right|\le \frac{d-c}{8}[{(|X\prime (d,.){|}^q+\frac{1}{4}\eta (|X\prime (c,.){|}^q,|X\prime (d,.){|}^q))}^{1/q}\hfill \\ \hspace{0.17em}\hfill & +{(|X\prime (d,.){|}^q+\frac{3}{4}\eta (|X\prime (c,.){|}^q,|X\prime (d,.){|}^q))}^{1/q}]\hspace{0.17em}\hspace{0.17em}(a.e.).\hfill \end{array}$$

Remark 9.

When $\mu (.)=0$ and $\eta (|X\prime (c,.){|}^q,|X\prime (d,.){|}^q)=|X\prime (c,.){|}^q-|X\prime (d,.){|}^q,$ then above theorem reduces to Corollary 5.13 of (Fu et al. 2021).

4. Conclusion

We have defined strong η-convexity for stochastic processes. We have established the Hermite-Hadamard, Ostrowski and some important inequalities for the above mentioned processes. The results obtained in this paper are the generalization of the previously known results. In the future, new integral inequalities for the other generalized convex stochastic processes can be obtained using similar methods in this study.

Acknowledgments

The authors are indebted to the anonymous reviewers for their valuable comments and remarks that helped to improve the presentation and quality of the manuscript.

Additional information

Funding

The publication of this article was funded by the Qatar National Library.