In this paper, we introduce and study the class of modified \((p, h)\)-convex stochastic processes, which unifies and extends several existing notions of convexity in the stochastic setting. We establish fundamental arithmetic properties of this class and derive Hermite–Hadamard-type inequalities using classical and fractional Katugampola-type operators. We also investigated Ostrowski-type and Jensen-type inequalities in a simple and unified framework. Our results generalize and unify many existing results in the literature, providing a comprehensive framework for the study of convex stochastic processes.
The theory of stochastic processes has been extensively developed and applied across various fields, including probability theory, optimization, mathematical finance, and engineering. Among the numerous classes of stochastic processes, convex stochastic processes have attracted significant attention due to their rich structural properties and wide range of applications. The concept of convex stochastic processes was first introduced by Nikodem [1], who established fundamental properties and initiated the systematic study of these processes. Since then, various generalizations and extensions have been explored by several researchers.
Skowronski [2] investigated properties of \(j\)-convex stochastic processes, while later, Skowronski [3] examined Wright-convex stochastic processes, contributing to the understanding of different convexity notions in the stochastic setting. The Hermite-Hadamard inequality, a classical result in convex analysis, was extended to convex stochastic processes by Kotrys [4], who provided significant insights into the integral inequalities satisfied by such processes. Subsequently, Kotrys [5] further developed the theory by studying strongly convex stochastic processes and establishing related inequalities.
The concept of \(h\)-convexity was introduced in the context of stochastic processes by Barrez et al. [6], who explored the properties and inequalities associated with \(h\)-convex stochastic processes. This work was later generalized by Saleem et al. [7], who provided broader characterizations and classical inequalities for \(h\)-convex stochastic processes.
In recent years, several new classes of convex stochastic processes have been introduced and studied. Okur, Işcan, and Dizdar [8] investigated \(p\)-convex stochastic processes and established Hermite-Hadamard type inequalities for this class. Maden, Tomar, and Set [9] studied \(s\)-convex stochastic processes in the first sense, while Set, Tomar, and Maden [10] examined \(s\)-convex stochastic processes in the second sense. Akdemir, Bekar, and Işcan [11] explored preinvexity for stochastic processes, extending the notion of invexity to the stochastic framework.
The study of exponentially convex and \(m\)-convex stochastic processes was undertaken by Özcan [12, 13], who derived Hermite-Hadamard type inequalities for these classes. More recently, Fu et al. [14] investigated \(n\)-polynomial convex stochastic processes and established related inequalities, further expanding the theory.
The theoretical developments in convex stochastic processes have found applications in various domains. Stochastic fractional calculus with applications to variational principles was explored by Zine and Torres [15]. The connections to optimization problems have been investigated by Chen, Quarteroni, and Rozza [16], who studied stochastic optimal Robin boundary control problems. Cuoco [17] and Cvitanic and Karatzas [18] examined applications in portfolio optimization and equilibrium prices with stochastic income. Xu and Yin [19] studied block stochastic gradient iteration methods for convex and nonconvex optimization problems. The foundational theory of stochastic differential equations and their applications to physics and engineering is comprehensively presented by Sobczyk [20]. Recent advancements in stochastic inequalities have focused on two interconnected generalizations: expanding convexity classes and employing interval-valued analysis. Afzal, Botmart, et al. [22] established foundational estimates for Jensen and Hermite-Hadamard inequalities within the class of h-Godunova-Levin stochastic processes. This framework was later extended to \((h_1, h_2)\)-convex stochastic processes by Afzal, Abbas, et al. [24], utilizing set inclusion relations.
Parallel to these developments, interval-valued methods have been employed to refine inequality estimates. Afzal, Prosviryakov, et al. [23] derived Hermite-Hadamard, Ostrowski, and Jensen-type inclusions for h-convex stochastic processes, while Abbas et al. [25] applied the center-radius (CR) order relation to obtain sharper bounds. Synthesizing these approaches, Afzal, Aloraini, et al. [21] introduced novel Kulisch-Miranker type inclusions for a generalized class of Godunova-Levin stochastic processes, representing a significant advance in computationally tractable interval-based inequalities.
The concept of modified \((p,h)\)-convexity [26], which combines the features of \(p\)-convex and \(h\)-convex functions, has recently gained attention. This class generalizes several existing notions of convexity and provides a unified framework for studying various types of convex functions and stochastic processes.
In this article, we aim to extend the theory of convex stochastic processes by introducing and studying the class of modified \((p,h)\)-convex stochastic processes. We establish fundamental properties, including closure under addition and scalar multiplication, and derive several important inequalities, such as Hermite-Hadamard type inequalities, Ostrowski-type inequalities, and Jensen-type inequalities. Our results generalize and unify many existing results in the literature, providing a comprehensive framework for the study of convex stochastic processes with applications in various fields.
The paper is organized as follows. In §2, we recall necessary definitions and preliminary results. §3 introduces the class of modified \((p,h)\)-convex stochastic processes and establishes their basic properties. §4 presents Hermite-Hadamard type inequalities and other integral inequalities for these processes. §5 provides concluding remarks and future directions.
In this section, we recall some fundamental concepts related to random variables and stochastic processes, together with their regularity properties and several relevant auxiliary results. For more details, we refer to [27].
Let \((\Pi, \mathcal{A}, \mathbb{P})\) be a probability space. A random variable is a measurable mapping \(\psi: \Pi \to \mathbb{R}\), that is, \(\psi\) is \(\mathcal{A}\)-measurable.
Definition 1. A stochastic process is a mapping \(\psi: J \times \Pi \to \mathbb{R}\), where \(J \subset \mathbb{R}\) is an index set. For each fixed \(t \in J\), the function \(\psi(t, \cdot): \Pi \to \mathbb{R}\) is a random variable. For each fixed \(\omega \in \Pi\), the mapping \(t \mapsto \psi(t, \omega)\) is called a sample path (or realization) of the process.
Let \(\psi: J \times \Pi \to \mathbb{R}\) be a stochastic process defined on \((\Pi, \mathcal{A}, \mathbb{P})\).
The process \(\psi\) is said to be continuous in probability at \(t_0 \in J\) if for every \(\varepsilon > 0\), \[\lim_{t \to t_0} \mathbb{P}\!\left( |\psi(t, \cdot) – \psi(t_0, \cdot)| > \varepsilon \right) = 0.\]
The process \(\psi\) is called mean square continuous at \(t_0 \in J\) if \[\lim_{t \to t_0} \mathbb{E}\!\left[ \left( \psi(t, \cdot) – \psi(t_0, \cdot) \right)^2 \right] = 0.\]
Mean square continuity implies continuity in probability, but the converse is not true.
The process \(\psi\) is said to be mean square differentiable at \(t_0 \in J\) if there exists a random variable \(\psi'(t_0, \cdot)\) such that \[\lim_{t \to t_0} \mathbb{E}\!\left[ \left( \frac{\psi(t, \cdot) – \psi(t_0, \cdot)}{t – t_0} – \psi'(t_0, \cdot) \right)^2 \right] = 0.\]
Suppose we are given a sequence \(\{\Delta^m\}\) of partitions, \(\Delta^m = \{t_{m,0}, t_{m,1}, \ldots, t_{m,n_m}\}\). We say that \(\{\Delta^m\}\) is a normal sequence of partitions if the length of the greatest interval in the \(m\)-th partition tends to zero, i.e., \[\lim_{m \to \infty} \sup_{1 \leq i \leq n_m} |t_{m,i} – t_{m,i-1}| = 0.\]
Assume that \(\mathbb{E}[\psi(t, \cdot)^2] < \infty\) for all \(t \in J\). Let \([l, m] \subset J\) and consider a partition \(l = t_0 < t_1 < \cdots < t_n = m\) with \(\xi_k \in [t_{k-1}, t_k]\) for \(k = 1, \ldots, n\). A random variable \(Y: \Pi \to \mathbb{R}\) is called the mean square integral of \(\psi\) over \([l, m]\) if \[\lim_{n \to \infty} \mathbb{E}\!\left[ \left( \sum_{k=1}^n \psi(\xi_k, \cdot)(t_k – t_{k-1}) – Y \right)^2 \right] = 0\] for every normal sequence of partitions of \([l, m]\). In this case, we write \[Y(\cdot) = \int_l^m \psi(s, \cdot) \, ds \quad \text{(a.e.)}.\]
For the existence of the mean-square integral, it suffices to assume the mean-square continuity of \(\psi\). Throughout the paper, we will frequently use the monotonicity property: if \(\psi(t, \cdot) \leq \phi(t, \cdot)\) (a.e.) on \([l, m]\), then \[\int_l^m \psi(t, \cdot) \, dt \leq \int_l^m \phi(t, \cdot) \, dt \quad \text{(a.e.)}.\]
Now, we present some results about Hermite-Hadamard inequality for convex stochastic processes.
Lemma 1. If \(\psi: J \times \Pi \to \mathbb{R}\) is a stochastic process of the form \(\psi(t, \cdot) = A(\cdot) t + B(\cdot)\), where \(A, B: \Pi \to \mathbb{R}\) are random variables such that \(\mathbb{E}[A^2] < \infty\) and \(\mathbb{E}[B^2] < \infty\), and \([l, m] \subset J\), then \[\int_l^m \psi(t, \cdot) \, dt = A(\cdot) \frac{m^2 – l^2}{2} + B(\cdot)(m – l) \quad \text{(a.e.)}.\]
Proposition 1. Let \(\psi: J \times \Pi \to \mathbb{R}\) be a convex stochastic process and \(t_0 \in \operatorname{int} J\). Then there exists a random variable \(A: \Pi \to \mathbb{R}\) such that \(\psi\) is supported at \(t_0\) by the process \(A(\cdot)(t – t_0) + \psi(t_0, \cdot)\). That is, \[\psi(t, \cdot) \geq A(\cdot)(t – t_0) + \psi(t_0, \cdot) \quad \text{(a.e.)},\] for all \(t \in J\).
Theorem 1(Hermite-Hadamard Inequality for Convex Stochastic Processes).Let \(\psi: J \times \Pi \to \mathbb{R}\) be a convex, mean-square continuous stochastic process. Then for any \(l, m \in J\) with \(l < m\), we have \[\psi\left(\frac{l + m}{2}, \cdot\right) \leq \frac{1}{m – l} \int_l^m \psi(t, \cdot) \, dt \leq \frac{\psi(l, \cdot) + \psi(m, \cdot)}{2} \quad \text{(a.e.)}.\]
Let \((\Pi, \mathcal{A}, \mathbb{P})\) be a probability space and let \(J \subset \mathbb{R}\) be an interval. A function \(\psi: J \times \Pi \to \mathbb{R}\) is a stochastic process if for every \(t \in J\) the function \(\psi(t, \cdot)\) is a random variable. All inequalities involving stochastic processes are understood to hold almost everywhere (a.e.).
Definition 2([1]).A stochastic process \(\psi: J \times \Pi \to \mathbb{R}\) is called a convex stochastic process if \[\psi(sm + (1-s)n, \cdot) \le s\,\psi(m, \cdot) + (1-s)\,\psi(n, \cdot), \tag{1}\] for all \(m, n \in J\) and \(s \in [0,1]\).
Definition 3([6]).Let \(h: (0,1) \to [0, \infty)\) be a given function. A non-negative stochastic process \(\psi: J \times \Pi \to \mathbb{R}\) is called an \(h\)-convex stochastic process if \[\psi(sm + (1-s)n, \cdot) \le h(s)\,\psi(m, \cdot) + h(1-s)\,\psi(n, \cdot), \tag{2}\] for all \(m, n \in J\) and \(s \in (0,1)\).
Definition 4([26]).The interval \(J \subset \mathbb{R}\) is called a \(p\)-convex set if for every \(m, n \in J\) and \(s \in [0,1]\), we have \[\left[ s m^p + (1-s) n^p \right]^{\frac{1}{p}} \in J,\] where \(p = \frac{n}{m}\) with \(n, m\) odd positive integers, or \(p = 2k+1\) for \(k \in \mathbb{N}\).
Definition 5([28]).Let \(p > 0\) and let \(J\) be a \(p\)-convex set. A non-negative stochastic process \(\psi: J \times \Pi \to \mathbb{R}\) is called a \(p\)-convex stochastic process if \[\psi\!\left(\bigl(s m^{p} + (1-s) n^{p}\bigr)^{1/p}, \cdot\right) \le s\,\psi(m, \cdot) + (1-s)\,\psi(n, \cdot), \tag{3}\] for all \(m, n \in J\) and \(s \in (0,1)\).
Definition 6([29]).Let \(p > 0\) and let \(h: (0,1) \to [0, \infty)\) be a non-zero function. Assume that \(J \subset \mathbb{R}\) is a \(p\)-convex set. A non-negative stochastic process \(\psi: J \times \Pi \to \mathbb{R}\) is called a \((p,h)\)-convex stochastic process if \[\psi\!\left(\bigl(s m^{p} + (1-s) n^{p}\bigr)^{1/p}, \cdot\right) \le h(s)\,\psi(m, \cdot) + h(1-s)\,\psi(n, \cdot), \tag{4}\] for all \(m, n \in J\) and \(s \in (0,1)\).
In this section, we first define the idea of a modified \((p, h)\)-convex stochastic process and, utilizing this, we investigate a few structural properties along with some inequalities.
Definition 7. Let \(p > 0\) and let \(h: J \subset \mathbb{R} \rightarrow \mathbb{R}\) be a non-negative and non-zero function. Assume \(J\) is a \(p\)-convex set. A non-negative stochastic process \(\psi: J \times \Pi \rightarrow \mathbb{R}\) is called a modified \((p,h)\)-convex stochastic process if, for every \(m, n \in J\) and \(s \in (0,1)\), the following inequality holds: \[\psi\!\left( \left[ s m^p + (1-s) n^p \right]^{\frac{1}{p}}, \cdot \right) \leq h(s)\,\psi(m, \cdot) + (1 – h(s))\,\psi(n, \cdot). \tag{5}\]
The modified \((p,h)\)-convex stochastic process defined above generalizes and unifies several existing notions of convex stochastic processes. Specifically, by appropriate choices of the parameters \(p\) and the function \(h\), we can recover the following classical definitions:
Classical convex stochastic process: Take \(p = 1\) and \(h(s) = s\) for all \(s \in (0,1)\). Then inequality (1) reduces to \[\psi(sm + (1-s)n, \cdot) \le s\,\psi(m, \cdot) + (1-s)\,\psi(n, \cdot),\] which is exactly the definition of a classical convex stochastic process [1].
\(p\)-convex stochastic process: Take \(h(s) = s\) for all \(s \in (0,1)\) while keeping \(p > 0\). Then (1) becomes \[\psi\!\left(\left[ s m^p + (1-s) n^p \right]^{\frac{1}{p}}, \cdot\right) \le s\,\psi(m, \cdot) + (1-s)\,\psi(n, \cdot),\] which is precisely the definition of a \(p\)-convex stochastic process [28].
Proposition 2.If \(h_1, h_2: J \subset \mathbb{R} \rightarrow \mathbb{R}\) are non-negative functions with \(h_2(s) \leq h_1(s)\) for all \(s \in (0,1)\) and \(\psi: J \times \Pi \rightarrow \mathbb{R}\) is a non-negative modified \((p, h_2)\)-convex stochastic process, then \(\psi\) is a modified \((p, h_1)\)-convex stochastic process.
Proof. Consider \(m, n \in J\) and \(s \in (0,1)\) arbitrary. Then, \[\begin{aligned} \psi\!\left( \left[ s m^p + (1-s) n^p \right]^{\frac{1}{p}}, \cdot \right) & \leq h_2(s)\,\psi(m, \cdot) + (1 – h_2(s))\,\psi(n, \cdot) \\ & \leq h_1(s)\,\psi(m, \cdot) + (1 – h_1(s))\,\psi(n, \cdot) \quad (\text{a.e.}). \end{aligned}\] ◻
Proposition 3. Let \(h: J \subset \mathbb{R} \rightarrow \mathbb{R}\) be a non-negative function. If \(\psi, \phi: J \times \Pi \rightarrow \mathbb{R}\) are modified \((p, h)\)-convex stochastic processes, then \(\psi + \phi\) is a modified \((p, h)\)-convex stochastic process. Also, if \(\alpha > 0\) then \(\alpha \psi\) is a modified \((p, h)\)-convex stochastic process.
Proof.Consider \(m, n \in J\) and \(s \in (0,1)\) arbitrary. \[\begin{aligned} (\psi + \phi) \left( \left[ s m^p + (1-s) n^p \right]^{\frac{1}{p}}, \cdot \right) & = \psi \left( \left[ s m^p + (1-s) n^p \right]^{\frac{1}{p}}, \cdot \right) + \phi \left( \left[ s m^p + (1-s) n^p \right]^{\frac{1}{p}}, \cdot \right) \\ & \leq h(s) \left( \psi(m, \cdot) + \phi(m, \cdot) \right) + (1 – h(s)) \left( \psi(n, \cdot) + \phi(n, \cdot) \right) \\ & = h(s) (\psi + \phi)(m, \cdot) + (1 – h(s)) (\psi + \phi)(n, \cdot) \quad (\text{a.e.}). \end{aligned}\]
Now, consider \(\alpha > 0\). Then, \[\begin{aligned} \alpha \psi \left( \left[ s m^p + (1-s) n^p \right]^{\frac{1}{p}}, \cdot \right) & \leq \alpha h(s) \psi(m, \cdot) + \alpha (1 – h(s)) \psi(n, \cdot) \\ & = h(s) \alpha \psi(m, \cdot) + (1 – h(s)) \alpha \psi(n, \cdot) \quad (\text{a.e.}) \end{aligned}\] ◻
Proposition 4. Let \(h_1, h_2: J \subset \mathbb{R} \rightarrow \mathbb{R}\) be non-negative functions and \(\psi, \phi: J \times \Pi \rightarrow \mathbb{R}\) non-negative stochastic processes such that \[\left(\psi(m, \cdot) – \psi(n, \cdot)\right)\left(\phi(m, \cdot) – \phi(n, \cdot)\right) \geq 0,\] for all \(m, n \in J\). If \(\psi\) is modified \((p, h_1)\)-convex, \(\phi\) is modified \((p, h_2)\)-convex and \(h(s) + (1 – h(s)) \leq c\) for all \(s \in (0,1)\), where \(h(s) = \max\{h_1(s), h_2(s)\}\) and \(c\) is a fixed positive number, then the product \(\psi \phi\) is a modified \((p, ch)\)-convex stochastic process.
Proof. Fix \(m, n \in J\) and \(s \in (0,1)\).
First, note that \(\left(\psi(m, \cdot) – \psi(n, \cdot)\right)\left(\phi(m, \cdot) – \phi(n, \cdot)\right) \geq 0\) implies \[\psi(m, \cdot) \phi(n, \cdot) + \psi(n, \cdot) \phi(m, \cdot) \leq \psi(m, \cdot) \phi(m, \cdot) + \psi(n, \cdot) \phi(n, \cdot).\]
Hence, \[\begin{aligned} \psi \phi & \left( \left[ s m^p + (1-s) n^p \right]^{\frac{1}{p}}, \cdot \right) \\ & \leq \left( h_1(s) \psi(m, \cdot) + (1 – h_1(s)) \psi(n, \cdot) \right) \left( h_2(s) \phi(m, \cdot) + (1 – h_2(s)) \phi(n, \cdot) \right) \\ & \leq \left( h(s) \psi(m, \cdot) + (1 – h(s)) \psi(n, \cdot) \right) \left( h(s) \phi(m, \cdot) + (1 – h(s)) \phi(n, \cdot) \right) \\ & \leq h^2(s) \psi \phi(m, \cdot) + h(s)(1 – h(s)) \psi \phi(m, \cdot) + h(s)(1 – h(s)) \psi \phi(n, \cdot) + (1 – h(s))^2 \psi \phi(n, \cdot) \\ & = \big( h(s) + (1 – h(s)) \big) \left( h(s) \psi \phi(m, \cdot) + (1 – h(s)) \psi \phi(n, \cdot) \right) \\ & \leq c \, h(s) (\psi \phi)(m, \cdot) + c \, (1 – h(s)) (\psi \phi)(n, \cdot) \quad (\text{a.e.}). \end{aligned}\] ◻
Theorem 2. Let \(J\) be a \(p\)-convex interval such that \(0 \in J\) and \(h: J \subset \mathbb{R} \rightarrow \mathbb{R}\) a non-negative function. If \(h\) is supermultiplicative and \(\psi: J \times \Pi \rightarrow \mathbb{R}\) is a modified \((p, h)\)-convex stochastic process such that \(\psi(0, \cdot) = 0\), then the inequality \[\psi\!\left( \left[ \lambda m^p + \beta n^p \right]^{\frac{1}{p}}, \cdot \right) \leq h(\lambda) \psi(m, \cdot) + (1 – h(\beta)) \psi(n, \cdot),\] holds almost everywhere for all \(m, n \in J\) and all \(\lambda, \beta > 0\) such that \(\lambda + \beta \leq 1\).
Proof. If \(\lambda + \beta = 1\), the inequality holds because of the definition of modified \((p, h)\)-convexity in stochastic processes. Let \(\lambda, \beta > 0\) be numbers such that \(\lambda + \beta = \gamma\) with \(\gamma < 1\). Let us define numbers \(a := \frac{\lambda}{\gamma}\) and \(b := \frac{\beta}{\gamma}\). Then, \(a + b = 1\) and we have the following: \[\begin{aligned} \psi\!\left( \left[ \lambda m^p + \beta n^p \right]^{\frac{1}{p}}, \cdot \right) &= \psi\!\left( \left[ a \gamma m^p + b \gamma n^p \right]^{\frac{1}{p}}, \cdot \right) \\ &\leq h(a) \psi\!\left( \left[ \gamma m^p + 0 \cdot n^p \right]^{\frac{1}{p}}, \cdot \right) + (1 – h(a)) \psi\!\left( \left[ \gamma n^p + 0 \cdot m^p \right]^{\frac{1}{p}}, \cdot \right) \\ &= h(a) \psi\!\left( \left[ \gamma m^p + (1-\gamma) 0^p \right]^{\frac{1}{p}}, \cdot \right) + (1 – h(a)) \psi\!\left( \left[ \gamma n^p + (1-\gamma) 0^p \right]^{\frac{1}{p}}, \cdot \right) \\ &\leq h(a) \left( h(\gamma) \psi(m, \cdot) + (1 – h(\gamma)) \psi(0, \cdot) \right) + (1 – h(a)) \left( h(\gamma) \psi(n, \cdot) + (1 – h(\gamma)) \psi(0, \cdot) \right) \\ &= h(a) h(\gamma) \psi(m, \cdot) + (1 – h(a)) h(\gamma) \psi(n, \cdot) \\ &\leq h(a \gamma) \psi(m, \cdot) + (1 – h(a \gamma)) \psi(n, \cdot) \\ &= h(\lambda) \psi(m, \cdot) + (1 – h(\lambda)) \psi(n, \cdot) \quad (\text{a.e.}). \end{aligned}\] ◻
Proposition 5. Let \(h: J \subset \mathbb{R} \rightarrow \mathbb{R}\) be a non-negative supermultiplicative function and let \(\psi: J \times \Pi \rightarrow \mathbb{R}\) be a modified \((p, h)\)-convex stochastic process. Then, for \(m, n, r \in J\), with \(m < n < r\) such that \(r – m, r – n, n – m \in J\) the following inequality holds almost everywhere, \[h(r – n) \psi(m, \cdot) – h(r – m) \psi(n, \cdot) + h(n – m) \psi(r, \cdot) \geq 0.\]
Proof. Let \(\psi: J \times \Pi \rightarrow \mathbb{R}\) be a modified \((p, h)\)-convex stochastic process and \(m, n, r \in J\) be numbers which satisfy assumptions of the proposition. Then, we have \(\frac{r – n}{r – m}, \frac{n – m}{r – m} \in (0,1)\) and \(\frac{r – n}{r – m} + \frac{n – m}{r – m} = 1\). Also, since \(h\) is supermultiplicative and non-negative, \[h(r – n) = h\left( \frac{r – n}{r – m} \cdot (r – m) \right) \geq h\left( \frac{r – n}{r – m} \right) h(r – m),\] \[h(n – m) = h\left( \frac{n – m}{r – m} \cdot (r – m) \right) \geq h\left( \frac{n – m}{r – m} \right) h(r – m).\]
Let \(h(r – m) > 0\). Because of the modified \((p, h)\)-convexity, \(\psi\) satisfies \[\psi\!\left( \left[ s z_1^p + (1-s) z_2^p \right]^{\frac{1}{p}}, \cdot \right) \leq h(s) \psi(z_1, \cdot) + (1 – h(s)) \psi(z_2, \cdot) \quad (\text{a.e.}),\] for all \(z_1, z_2 \in J, s \in (0,1)\). In particular, for \(s = \frac{r – n}{r – m}\), \(z_1 = m\), \(z_2 = r\), we have \(n = \left[ s m^p + (1-s) r^p \right]^{\frac{1}{p}}\) and \[\begin{aligned} \psi(n, \cdot) & \leq h\left( \frac{r – n}{r – m} \right) \psi(m, \cdot) + \left(1 – h\left( \frac{r – n}{r – m} \right)\right) \psi(r, \cdot) \\ & \leq \frac{h(r – n)}{h(r – m)} \psi(m, \cdot) + \left(1 – \frac{h(r – n)}{h(r – m)}\right) \psi(r, \cdot). \end{aligned}\]
Finally, multiplying by \(h(r – m)\) we obtain the following \[h(r – m) \psi(n, \cdot) \leq h(r – n) \psi(m, \cdot) + \left( h(r – m) – h(r – n) \right) \psi(r, \cdot).\]
That is, \[0 \leq h(r – n) \psi(m, \cdot) – h(r – m) \psi(n, \cdot) + \left( h(r – m) – h(r – n) \right) \psi(r, \cdot).\]
Note: The expression simplifies to the desired form only when \(h(r – m) – h(r – n) = h(n – m)\), which holds under the supermultiplicative property and the condition \(h(r – m) = h(r – n + n – m) \geq h(r – n) h(n – m)\). ◻
The equality stated in the following lemma is fundamental for proving the results related to Ostrowski-type inequalities. We mainly extend the results for the \(h\)-convex stochastic process (Lemma 2.1) in [30] to the case of the modified \((p,h)\)-convex stochastic process.
Lemma 2. Let \(\psi: J \times \Pi \rightarrow \mathbb{R}\) be a stochastic process mean-square differentiable on \(J^{\circ}\). If \(\psi^{\prime}\) is mean-square integrable on \([a, b]\), where \(a, b \in J\) with \(a < b\), then the following equality holds: \[\begin{aligned} \psi(t, \cdot) – \frac{1}{b-a} \int_a^b \psi(u, \cdot) \, du = & \frac{(t-a)^2}{b-a} \int_0^1 s \, \psi^{\prime}\left( \left[ s t^p + (1-s) a^p \right]^{\frac{1}{p}}, \cdot \right) ds \\ & – \frac{(b-t)^2}{b-a} \int_0^1 s \, \psi^{\prime}\left( \left[ s t^p + (1-s) b^p \right]^{\frac{1}{p}}, \cdot \right) ds, \end{aligned}\] for each \(t \in [a, b]\).
Proof. Changing variables, \(u = \left[ s t^p + (1-s) a^p \right]^{\frac{1}{p}}\) and \(w = \left[ s t^p + (1-s) b^p \right]^{\frac{1}{p}}\) and integrating by parts, we have \[\begin{aligned} \frac{(t-a)^2}{b-a} &\int_0^1 s \, \psi^{\prime}\left( \left[ s t^p + (1-s) a^p \right]^{\frac{1}{p}}, \cdot \right) ds – \frac{(b-t)^2}{b-a} \int_0^1 s \, \psi^{\prime}\left( \left[ s t^p + (1-s) b^p \right]^{\frac{1}{p}}, \cdot \right) ds \\ = & \frac{(t-a)^2}{b-a} \int_a^t \frac{(u-a)}{(t-a)} \psi^{\prime}(u, \cdot) \frac{du}{(t-a)} – \frac{(b-t)^2}{b-a} \int_t^b \frac{(b-w)}{(b-t)} \psi^{\prime}(w, \cdot) \frac{dw}{(b-t)} \\ = & \frac{1}{b-a} \int_a^t (u-a) \psi^{\prime}(u, \cdot) \, du – \frac{1}{b-a} \int_t^b (b-w) \psi^{\prime}(w, \cdot) \, dw \\ = & \frac{1}{b-a} \left[ (t-a) \psi(t, \cdot) – \int_a^t \psi(u, \cdot) \, du \right] + \frac{1}{b-a} \left[ (b-t) \psi(t, \cdot) – \int_t^b \psi(w, \cdot) \, dw \right] \\ = & \frac{1}{b-a} \left[ (t-a) \psi(t, \cdot) – \int_a^t \psi(u, \cdot) \, du + (b-t) \psi(t, \cdot) – \int_t^b \psi(w, \cdot) \, dw \right] \\ = & \frac{1}{b-a} \left[ \psi(t, \cdot) (b-a) – \int_a^b \psi(u, \cdot) \, du \right]. \end{aligned}\] ◻
Now, we are ready to present some Ostrowski-type inequalities for the modified \((p,h)\)-convex stochastic process. Mainly, we improve Theorem 2.2 in [30].
Theorem 3. Let \(h: J \subset \mathbb{R} \rightarrow \mathbb{R}\) be a non-negative and supermultiplicative function such that \(h(\alpha) > \alpha\) for every \(\alpha\) and let \(\psi: J \times \Pi \rightarrow \mathbb{R}\) be a mean-square stochastic process such that \(\psi^{\prime}\) is mean-square integrable on \([a, b]\), where \(a, b \in J\) with \(a < b\). If \(|\psi^{\prime}|\) is a modified \((p, h)\)-convex stochastic process on \(J\) and \(|\psi^{\prime}(t, \cdot)| \leq M\) for every \(t\), then \[\begin{aligned} & \left| \psi(t, \cdot) – \frac{1}{b-a} \int_a^b \psi(u, \cdot) \, du \right| \leq \frac{M \left[ (t-a)^2 + (b-t)^2 \right]}{b-a} \cdot \frac{1}{2}. \end{aligned}\] for each \(t \in [a, b]\).
Proof. By Lemma 2 and since \(|\psi^{\prime}|\) is modified \((p, h)\)-convex, then we can write \[\begin{aligned} \left| \psi(t, \cdot) – \frac{1}{b-a} \int_a^b \psi(u, \cdot) \, du \right| \leq & \frac{(t-a)^2}{b-a} \int_0^1 s \left| \psi^{\prime} \left( \left[ s t^p + (1-s) a^p \right]^{\frac{1}{p}}, \cdot \right) \right| ds \\ & + \frac{(b-t)^2}{b-a} \int_0^1 s \left| \psi^{\prime} \left( \left[ s t^p + (1-s) b^p \right]^{\frac{1}{p}}, \cdot \right) \right| ds \\ \leq & \frac{(t-a)^2}{b-a} \int_0^1 s \left[ h(s) \left| \psi^{\prime}(t, \cdot) \right| + (1 – h(s)) \left| \psi^{\prime}(a, \cdot) \right| \right] ds \\ & + \frac{(b-t)^2}{b-a} \int_0^1 s \left[ h(s) \left| \psi^{\prime}(t, \cdot) \right| + (1 – h(s)) \left| \psi^{\prime}(b, \cdot) \right| \right] ds \\ \leq & \frac{M (t-a)^2}{b-a} \int_0^1 \left[ s h(s) + s (1 – h(s)) \right] ds + \frac{M (b-t)^2}{b-a} \int_0^1 \left[ s h(s) + s (1 – h(s)) \right] ds \\ = & \frac{M (t-a)^2}{b-a} \int_0^1 s \, ds + \frac{M (b-t)^2}{b-a} \int_0^1 s \, ds \\ = & \frac{M \left[ (t-a)^2 + (b-t)^2 \right]}{b-a} \cdot \frac{1}{2}. \end{aligned}\] ◻
Now we prove a Jensen-type inequality for the modified \((p,h)\)-convex stochastic process.
Theorem 4. Assume \(\psi: J \times \Pi \rightarrow \mathbb{R}\) be a modified \((p, h)\)-convex stochastic process and \(h\) be non-negative supermultiplicative function. If \(T_i = \sum_{j=1}^i \alpha_j\) for \(i = 1, \ldots, m\), so that \(T_m = 1\) where \(m \in \mathbb{N}\), then \[\psi\!\left( \left( \sum_{i=1}^m \alpha_i r_i^p \right)^{\frac{1}{p}}, \cdot \right) \leq \psi(r_m, \cdot) + \sum_{i=1}^{m-1} h(T_i) \left\{ \psi(r_i, \cdot) – \psi(r_{i+1}, \cdot) \right\}.\]
Proof. By the definition of modified \((p, h)\)-convex stochastic process it follows that: \[\begin{aligned} \psi\!\left( \left( \sum_{i=1}^m \alpha_i r_i^p \right)^{\frac{1}{p}}, \cdot \right) & \leq h(T_{m-1}) \, \psi\!\left( \left( \sum_{i=1}^{m-1} \frac{\alpha_i}{T_{m-1}} r_i^p \right)^{\frac{1}{p}}, \cdot \right) + \left(1 – h(T_{m-1})\right) \psi(r_m, \cdot) \\ & = \left(1 – h(T_{m-1})\right) \psi(r_m, \cdot) + h(T_{m-1}) \, \psi\!\left( \left[ \frac{T_{m-2}}{T_{m-1}} \left( \left( \sum_{i=1}^{m-2} \frac{\alpha_i}{T_{m-2}} r_i^p \right)^{\frac{1}{p}} \right)^p + \frac{\alpha_{m-1}}{T_{m-1}} r_{m-1}^p \right]^{\frac{1}{p}}, \cdot \right) \\ & \leq \left(1 – h(T_{m-1})\right) \psi(r_m, \cdot) + h(T_{m-1}) \left[ h\!\left( \frac{T_{m-2}}{T_{m-1}} \right) \psi\!\left( \left( \sum_{i=1}^{m-2} \frac{\alpha_i}{T_{m-2}} r_i^p \right)^{\frac{1}{p}}, \cdot \right) \right. \\ & \quad \left. + \left(1 – h\!\left( \frac{T_{m-2}}{T_{m-1}} \right) \right) \psi(r_{m-1}, \cdot) \right]. \end{aligned}\]
Using the fact that \(h\) is supermultiplicative, we have
\[\begin{aligned} & \leq \left(1 – h(T_{m-1})\right) \psi(r_m, \cdot) + h(T_{m-2}) \, \psi\!\left( \left( \sum_{i=1}^{m-2} \frac{\alpha_i}{T_{m-2}} r_i^p \right)^{\frac{1}{p}}, \cdot \right) + h(T_{m-1}) \left(1 – h\!\left( \frac{T_{m-2}}{T_{m-1}} \right) \right) \psi(r_{m-1}, \cdot) \\ & = \psi(r_m, \cdot) + h(T_{m-1}) \psi(r_{m-1}, \cdot) – h(T_{m-1}) \psi(r_m, \cdot) – h(T_{m-2}) \psi(r_{m-1}, \cdot) \\ & \quad + h(T_{m-2}) \, \psi\!\left( \left( \sum_{i=1}^{m-2} \frac{\alpha_i}{T_{m-2}} r_i^p \right)^{\frac{1}{p}}, \cdot \right) \\ & = \psi(r_m, \cdot) + h(T_{m-1}) \left[ \psi(r_{m-1}, \cdot) – \psi(r_m, \cdot) \right] – h(T_{m-2}) \psi(r_{m-1}, \cdot) \\ & \quad + h(T_{m-2}) \, \psi\!\left( \left[ \frac{T_{m-3}}{T_{m-2}} \left( \left( \sum_{i=1}^{m-3} \frac{\alpha_i}{T_{m-3}} r_i^p \right)^{\frac{1}{p}} \right)^p + \frac{\alpha_{m-2}}{T_{m-2}} r_{m-2}^p \right]^{\frac{1}{p}}, \cdot \right). \end{aligned}\]
From the definition of modified \((p, h)\)-convexity, we get \[\begin{aligned} & \leq \psi(r_m, \cdot) + h(T_{m-1}) \left[ \psi(r_{m-1}, \cdot) – \psi(r_m, \cdot) \right] – h(T_{m-2}) \psi(r_{m-1}, \cdot) \\ & \quad + h(T_{m-2}) \left[ h\!\left( \frac{T_{m-3}}{T_{m-2}} \right) \psi\!\left( \left( \sum_{i=1}^{m-3} \frac{\alpha_i}{T_{m-3}} r_i^p \right)^{\frac{1}{p}}, \cdot \right) + \left(1 – h\!\left( \frac{T_{m-3}}{T_{m-2}} \right) \right) \psi(r_{m-2}, \cdot) \right]. \end{aligned}\]
Using the fact that \(h\) is supermultiplicative, \[\begin{aligned} & \leq \psi(r_m, \cdot) + h(T_{m-1}) \left[ \psi(r_{m-1}, \cdot) – \psi(r_m, \cdot) \right] – h(T_{m-2}) \psi(r_{m-1}, \cdot) \\ & \quad + h(T_{m-3}) \, \psi\!\left( \left( \sum_{i=1}^{m-3} \frac{\alpha_i}{T_{m-3}} r_i^p \right)^{\frac{1}{p}}, \cdot \right) + h(T_{m-2}) \psi(r_{m-2}, \cdot) – h(T_{m-3}) \psi(r_{m-2}, \cdot) \\ & = \psi(r_m, \cdot) + h(T_{m-1}) \left[ \psi(r_{m-1}, \cdot) – \psi(r_m, \cdot) \right] + h(T_{m-2}) \left[ \psi(r_{m-2}, \cdot) – \psi(r_{m-1}, \cdot) \right] \\ & \quad – h(T_{m-3}) \psi(r_{m-2}, \cdot) + h(T_{m-3}) \, \psi\!\left( \left( \sum_{i=1}^{m-3} \frac{\alpha_i}{T_{m-3}} r_i^p \right)^{\frac{1}{p}}, \cdot \right) \\ & \leq \ldots \\ & \leq \psi(r_m, \cdot) + h(T_{m-1}) \left[ \psi(r_{m-1}, \cdot) – \psi(r_m, \cdot) \right] + h(T_{m-2}) \left[ \psi(r_{m-2}, \cdot) – \psi(r_{m-1}, \cdot) \right] \\ & \quad + h(T_{m-3}) \left[ \psi(r_{m-3}, \cdot) – \psi(r_{m-2}, \cdot) \right] + \ldots \\ & \quad + h(T_2) \left[ \psi(r_2, \cdot) – \psi(r_3, \cdot) \right] + h(T_1) \left[ \psi(r_1, \cdot) – \psi(r_2, \cdot) \right] \\ & = \psi(r_m, \cdot) + \sum_{i=1}^{m-1} h(T_i) \left[ \psi(r_i, \cdot) – \psi(r_{i+1}, \cdot) \right]. \end{aligned}\]
The result is completed. ◻
Corollary 1. If we take \(p = 1\) and \(h(s) = s\) in Theorem 4, we obtain the Jensen-type inequality for the classical convex stochastic process: \[\psi\!\left( \sum_{i=1}^m \alpha_i r_i, \cdot \right) \leq \psi(r_m, \cdot) + \sum_{i=1}^{m-1} T_i \left\{ \psi(r_i, \cdot) – \psi(r_{i+1}, \cdot) \right\},\] where \(T_i = \sum_{j=1}^i \alpha_j\) with \(T_m = 1\) and \(m \in \mathbb{N}\).
Corollary 2. For the \(p\)-convex stochastic process, set \(h(s) = s\) in Theorem 4. Then we have: \[\psi\!\left( \left( \sum_{i=1}^m \alpha_i r_i^p \right)^{\frac{1}{p}}, \cdot \right) \leq \psi(r_m, \cdot) + \sum_{i=1}^{m-1} T_i \left\{ \psi(r_i, \cdot) – \psi(r_{i+1}, \cdot) \right\},\] where \(T_i = \sum_{j=1}^i \alpha_j\) with \(T_m = 1\) and \(m \in \mathbb{N}\).
We have now established the Hermite–Hadamard inequality for modified \((p, h)\)-convex stochastic processes, utilizing classical and fractional integral operators.
Theorem 5. Assume \(\psi: J \times \Pi \rightarrow \mathbb{R}\) be a modified \((p, h)\)-convex stochastic process on the interval \([l, m]\) with \(l < m\). Then we have \[\begin{aligned} \psi\left(\left(\frac{l^p + m^p}{2}\right)^{\frac{1}{p}}, \cdot \right) &\leq \left(\frac{p}{m^p – l^p}\right) \int_{l}^{m} r^{p-1} \psi(r, \cdot) \, dr \\ &\leq \psi(l, \cdot) + \left\{ \psi(m, \cdot) – \psi(l, \cdot) \right\} \int_0^1 h(s) \, ds. \end{aligned}\]
Proof. Let \(u^p = s l^p + (1-s) m^p\) and \(v^p = (1-s) l^p + s m^p\). Then we get \[\begin{aligned} \psi\left(\left(\frac{l^p + m^p}{2}\right)^{\frac{1}{p}}, \cdot \right) &= \psi\left(\left(\frac{u^p + v^p}{2}\right)^{\frac{1}{p}}, \cdot \right) \\ &= \psi\left(\left(\frac{(s l^p + (1-s) m^p) + ((1-s) l^p + s m^p)}{2}\right)^{\frac{1}{p}}, \cdot \right) \\ &\leq h\left(\frac{1}{2}\right) \psi\left((s l^p + (1-s) m^p)^{\frac{1}{p}}, \cdot \right) + \left(1 – h\left(\frac{1}{2}\right)\right) \psi\left(((1-s) l^p + s m^p)^{\frac{1}{p}}, \cdot \right). \end{aligned}\]
Integrating the inequality over \(s \in [0,1]\), we get \[\begin{aligned} \psi\left(\left(\frac{l^p + m^p}{2}\right)^{\frac{1}{p}}, \cdot \right) &\leq h\left(\frac{1}{2}\right) \int_0^1 \psi\left((s l^p + (1-s) m^p)^{\frac{1}{p}}, \cdot \right) ds + \left(1 – h\left(\frac{1}{2}\right)\right) \int_0^1 \psi\left(((1-s) l^p + s m^p)^{\frac{1}{p}}, \cdot \right) ds \\ &\leq h\left(\frac{1}{2}\right) \frac{p}{m^p – l^p} \int_{l}^{m} r^{p-1} \psi(r, \cdot) \, dr + \frac{p}{m^p – l^p} \left(1 – h\left(\frac{1}{2}\right)\right) \int_{l}^{m} r^{p-1} \psi(r, \cdot) \, dr \\ &= \frac{p}{m^p – l^p} \int_{l}^{m} r^{p-1} \psi(r, \cdot) \, dr. \end{aligned}\]
Now, we know that \[\begin{aligned} \int_{l}^{m} r^{p-1} \psi(r, \cdot) \, dr &= \frac{m^p – l^p}{p} \int_0^1 \psi\left((s m^p + (1-s) l^p)^{\frac{1}{p}}, \cdot \right) ds \\ &\leq \frac{m^p – l^p}{p} \int_0^1 \left[ h(s) \psi(m, \cdot) + (1 – h(s)) \psi(l, \cdot) \right] ds \\ &= \frac{m^p – l^p}{p} \left[ \psi(l, \cdot) + \left\{ \psi(m, \cdot) – \psi(l, \cdot) \right\} \int_0^1 h(s) \, ds \right]. \end{aligned}\]
Thus, \[\frac{p}{m^p – l^p} \int_{l}^{m} r^{p-1} \psi(r, \cdot) \, dr \leq \psi(l, \cdot) + \left\{ \psi(m, \cdot) – \psi(l, \cdot) \right\} \int_0^1 h(s) \, ds.\]
Combining the two inequalities, we obtain the required result. ◻
Now, before developing the fractional variant, we first extend the classical Katugampola operator [31] into its mean square stochastic form.
Definition 8(Katugampola Fractional Integral for Stochastic Processes). Let \(\psi: J \times \Pi \rightarrow \mathbb{R}\) be a stochastic process. Then, the mean-square continuous Katugampola fractional integrals \({}^{\rho}J_{l^+}^{\alpha} \psi\) and \({}^{\rho}J_{m^-}^{\alpha} \psi\) of order \(\alpha > 0\) and \(\rho > 0\) are defined by \[{}^{\rho}J_{l^+}^{\alpha} \psi(x, \cdot) := \frac{\rho^{1-\alpha}}{\Gamma(\alpha)} \int_l^x (x^{\rho} – t^{\rho})^{\alpha-1} t^{\rho-1} \psi(t, \cdot) \, dt, \quad \text{(a.e.)} \quad (0 \leq l < x < m),\] and \[{}^{\rho}J_{m^-}^{\alpha} \psi(x, \cdot) := \frac{\rho^{1-\alpha}}{\Gamma(\alpha)} \int_x^m (t^{\rho} – x^{\rho})^{\alpha-1} t^{\rho-1} \psi(t, \cdot) \, dt, \quad \text{(a.e.)} \quad (0 \leq l < x < m),\] where \(\Gamma(\alpha) = \int_0^{\infty} e^{-u} u^{\alpha-1} du\) is the Gamma function.
Theorem 6. Assume \(\psi: J \times \Pi \rightarrow \mathbb{R}\) be a modified \((p, h)\)-convex stochastic process and \(\psi \in L_1[l, m]\), with \(l < m\) and \(p, \rho > 0\). Then \[\frac{1}{\alpha} \psi\left(\left(\frac{l^p + m^p}{2}\right)^{\frac{1}{p}}, \cdot \right) \leq \frac{\rho^{\alpha-1} \Gamma(\alpha)}{(m^p – l^p)^{\alpha}} \left[ \left(1 – h\left(\frac{1}{2}\right)\right) {}^{\rho}J_{l^+}^{\alpha} \psi(m, \cdot) + h\left(\frac{1}{2}\right) {}^{\rho}J_{m^-}^{\alpha} \psi(l, \cdot) \right],\] and \[\frac{\rho^{\alpha-1} \Gamma(\alpha)}{(m^p – l^p)^{\alpha}} \left[ {}^{\rho}J_{l^+}^{\alpha} \psi(m, \cdot) + {}^{\rho}J_{m^-}^{\alpha} \psi(l, \cdot) \right] \leq \frac{\psi(l, \cdot) + \psi(m, \cdot)}{\alpha}.\]
Proof. We know that \(\psi\) is a modified \((p, h)\)-convex stochastic process, so we have \[\psi\left(\left(\frac{r^p + s^p}{2}\right)^{\frac{1}{p}}, \cdot \right) \leq \left\{1 – h\left(\frac{1}{2}\right)\right\} \psi(r, \cdot) + h\left(\frac{1}{2}\right) \psi(s, \cdot).\]
Let \(r^p = \left(t l^p + (1-t) m^p\right)\) and \(s^p = \left((1-t) l^p + t m^p\right)\). Then \[\psi\left(\left(\frac{l^p + m^p}{2}\right)^{\frac{1}{p}}, \cdot \right) \leq \left\{1 – h\left(\frac{1}{2}\right)\right\} \psi\left(\left(t l^p + (1-t) m^p\right)^{\frac{1}{p}}, \cdot \right) + h\left(\frac{1}{2}\right) \psi\left(\left((1-t) l^p + t m^p\right)^{\frac{1}{p}}, \cdot \right).\]
Multiplying the above inequality by \(t^{\alpha-1}\), then integrating over \(t \in [0,1]\), we get \[\begin{aligned} \frac{1}{\alpha} \psi\left(\left(\frac{l^p + m^p}{2}\right)^{\frac{1}{p}}, \cdot \right) = \psi\left(\left(\frac{l^p + m^p}{2}\right)^{\frac{1}{p}}, \cdot \right) \int_0^1 t^{\alpha-1} dt & \leq \left\{1 – h\left(\frac{1}{2}\right)\right\} \int_0^1 t^{\alpha-1} \psi\left(\left(t l^p + (1-t) m^p\right)^{\frac{1}{p}}, \cdot \right) dt \\ & \quad + h\left(\frac{1}{2}\right) \int_0^1 t^{\alpha-1} \psi\left(\left((1-t) l^p + t m^p\right)^{\frac{1}{p}}, \cdot \right) dt. \end{aligned}\]
Now, using the substitution \(u^p = t l^p + (1-t) m^p\), we have \(u \in [l, m]\). Then \[du = \frac{l^p – m^p}{p u^{p-1}} dt = -\frac{m^p – l^p}{p u^{p-1}} dt,\] so \(dt = \frac{p u^{p-1}}{m^p – l^p} du\). Also, note that \(t = \frac{m^p – u^p}{m^p – l^p}\) and \(t^{\alpha-1} = \left(\frac{m^p – u^p}{m^p – l^p}\right)^{\alpha-1}\). Thus, \[\begin{aligned} \int_0^1 t^{\alpha-1} \psi\left(\left(t l^p + (1-t) m^p\right)^{\frac{1}{p}}, \cdot \right) dt &= \int_l^m \left(\frac{m^p – u^p}{m^p – l^p}\right)^{\alpha-1} \psi(u, \cdot) \frac{p u^{p-1}}{m^p – l^p} du. \end{aligned}\]
Similarly, using the substitution \(v^p = (1-t) l^p + t m^p\), we have \(v \in [l, m]\), \(t = \frac{v^p – l^p}{m^p – l^p}\), and \[\int_0^1 t^{\alpha-1} \psi\left(\left((1-t) l^p + t m^p\right)^{\frac{1}{p}}, \cdot \right) dt = \int_l^m \left(\frac{v^p – l^p}{m^p – l^p}\right)^{\alpha-1} \psi(v, \cdot) \frac{p v^{p-1}}{m^p – l^p} dv.\]
Now, using the Katugampola fractional integral definition with parameter \(\rho\), we have \[{}^{\rho}J_{l^+}^{\alpha} \psi(m, \cdot) = \frac{\rho^{1-\alpha}}{\Gamma(\alpha)} \int_l^m (m^{\rho} – u^{\rho})^{\alpha-1} u^{\rho-1} \psi(u, \cdot) du,\] and \[{}^{\rho}J_{m^-}^{\alpha} \psi(l, \cdot) = \frac{\rho^{1-\alpha}}{\Gamma(\alpha)} \int_l^m (v^{\rho} – l^{\rho})^{\alpha-1} v^{\rho-1} \psi(v, \cdot) dv.\]
To connect these with our integrals, we set \(\rho = p\). Then, \[\begin{aligned} \int_l^m \left(\frac{m^p – u^p}{m^p – l^p}\right)^{\alpha-1} \psi(u, \cdot) \frac{p u^{p-1}}{m^p – l^p} du &= \frac{p}{(m^p – l^p)^{\alpha}} \int_l^m (m^p – u^p)^{\alpha-1} u^{p-1} \psi(u, \cdot) du \\ &= \frac{p}{(m^p – l^p)^{\alpha}} \cdot \frac{\Gamma(\alpha)}{p^{1-\alpha}} {}^{p}J_{l^+}^{\alpha} \psi(m, \cdot) \\ &= \frac{p^{\alpha} \Gamma(\alpha)}{(m^p – l^p)^{\alpha}} {}^{p}J_{l^+}^{\alpha} \psi(m, \cdot). \end{aligned}\]
Similarly, \[\int_l^m \left(\frac{v^p – l^p}{m^p – l^p}\right)^{\alpha-1} \psi(v, \cdot) \frac{p v^{p-1}}{m^p – l^p} dv = \frac{p^{\alpha} \Gamma(\alpha)}{(m^p – l^p)^{\alpha}} {}^{p}J_{m^-}^{\alpha} \psi(l, \cdot).\]
Substituting back, we obtain \[\begin{aligned} \frac{1}{\alpha} \psi\left(\left(\frac{l^p + m^p}{2}\right)^{\frac{1}{p}}, \cdot \right) &\leq \left\{1 – h\left(\frac{1}{2}\right)\right\} \frac{p^{\alpha} \Gamma(\alpha)}{(m^p – l^p)^{\alpha}} {}^{p}J_{l^+}^{\alpha} \psi(m, \cdot) + h\left(\frac{1}{2}\right) \frac{p^{\alpha} \Gamma(\alpha)}{(m^p – l^p)^{\alpha}} {}^{p}J_{m^-}^{\alpha} \psi(l, \cdot) \\ &= \frac{p^{\alpha} \Gamma(\alpha)}{(m^p – l^p)^{\alpha}} \left[ \left(1 – h\left(\frac{1}{2}\right)\right) {}^{p}J_{l^+}^{\alpha} \psi(m, \cdot) + h\left(\frac{1}{2}\right) {}^{p}J_{m^-}^{\alpha} \psi(l, \cdot) \right]. \end{aligned}\]
This proves the first inequality.
Now, we know that \(\psi\) is a modified \((p, h)\)-convex stochastic process, then \[\begin{aligned} & \psi\left(\left(t l^p + (1-t) m^p\right)^{\frac{1}{p}}, \cdot \right) + \psi\left(\left((1-t) l^p + t m^p\right)^{\frac{1}{p}}, \cdot \right) \\ & \leq h(t) \psi(l, \cdot) + (1 – h(t)) \psi(m, \cdot) + (1 – h(t)) \psi(l, \cdot) + h(t) \psi(m, \cdot) \\ & = \psi(l, \cdot) + \psi(m, \cdot). \end{aligned}\]
Multiplying the above inequality by \(t^{\alpha-1}\) and then integrating over \(t \in [0,1]\), we get \[\begin{aligned} \int_0^1 t^{\alpha-1} \psi\left(\left(t l^p + (1-t) m^p\right)^{\frac{1}{p}}, \cdot \right) dt & + \int_0^1 t^{\alpha-1} \psi\left(\left((1-t) l^p + t m^p\right)^{\frac{1}{p}}, \cdot \right) dt \leq \left[ \psi(l, \cdot) + \psi(m, \cdot) \right] \int_0^1 t^{\alpha-1} dt. \end{aligned}\]
Now, using the substitution \(u^p = t l^p + (1-t) m^p\) in the first integral, we have \(u \in [l, m]\), \(t = \frac{m^p – u^p}{m^p – l^p}\), and \(dt = \frac{p u^{p-1}}{m^p – l^p} du\). Thus, \[\int_0^1 t^{\alpha-1} \psi\left(\left(t l^p + (1-t) m^p\right)^{\frac{1}{p}}, \cdot \right) dt = \int_l^m \left(\frac{m^p – u^p}{m^p – l^p}\right)^{\alpha-1} \psi(u, \cdot) \frac{p u^{p-1}}{m^p – l^p} du.\]
Similarly, using the substitution \(v^p = (1-t) l^p + t m^p\) in the second integral, we have \(v \in [l, m]\), \(t = \frac{v^p – l^p}{m^p – l^p}\), and \(dt = \frac{p v^{p-1}}{m^p – l^p} dv\). Thus, \[\int_0^1 t^{\alpha-1} \psi\left(\left((1-t) l^p + t m^p\right)^{\frac{1}{p}}, \cdot \right) dt = \int_l^m \left(\frac{v^p – l^p}{m^p – l^p}\right)^{\alpha-1} \psi(v, \cdot) \frac{p v^{p-1}}{m^p – l^p} dv.\]
Now, using the Katugampola fractional integral definition with \(\rho = p\), we have
\[\int_l^m \left(\frac{m^p – u^p}{m^p – l^p}\right)^{\alpha-1} \psi(u, \cdot) \frac{p u^{p-1}}{m^p – l^p} du = \frac{p}{(m^p – l^p)^{\alpha}} \int_l^m (m^p – u^p)^{\alpha-1} u^{p-1} \psi(u, \cdot) du = \frac{p^{\alpha} \Gamma(\alpha)}{(m^p – l^p)^{\alpha}} {}^{p}J_{l^+}^{\alpha} \psi(m, \cdot),\] and \[\int_l^m \left(\frac{v^p – l^p}{m^p – l^p}\right)^{\alpha-1} \psi(v, \cdot) \frac{p v^{p-1}}{m^p – l^p} dv = \frac{p^{\alpha} \Gamma(\alpha)}{(m^p – l^p)^{\alpha}} {}^{p}J_{m^-}^{\alpha} \psi(l, \cdot).\]
Substituting these into the inequality, we obtain \[\frac{p^{\alpha} \Gamma(\alpha)}{(m^p – l^p)^{\alpha}} \left[ {}^{p}J_{l^+}^{\alpha} \psi(m, \cdot) + {}^{p}J_{m^-}^{\alpha} \psi(l, \cdot) \right] \leq \frac{\psi(l, \cdot) + \psi(m, \cdot)}{\alpha}.\] ◻
Remark 1.The results presented above are new, as they likely refine and improve the corresponding results in the deterministic case for the Riemann–Liouville operator [26]. Specifically, Theorem 3.2 is obtained in the deterministic sense by setting \(\rho = 1\).
In this paper, we introduced and studied the class of modified \((p,h)\)-convex stochastic processes, which unifies and generalizes several existing notions of convexity in the stochastic setting. We established fundamental properties of this class, including closure under addition and scalar multiplication. We derived several important inequalities, including Hermite-Hadamard type inequalities, Ostrowski-type inequalities, and Jensen-type inequalities. Our results provide a comprehensive framework for the study of convex stochastic processes and their applications.
Future research directions include extending these results to multidimensional stochastic processes, developing applications in stochastic optimization and financial mathematics, and investigating connections with other generalized convexity notions.
Nikodem, K. (1980). On convex stochastic processes. Aequationes Mathematicae, 20(1), 184-197.
Skowroński, A. (1992). On some properties of J-convex stochastic processes. Aequationes Mathematicae, 44(2), 249-258.
Skowroński, A. (1995, September). On Wright-convex stochastic processes. In Annales Mathematicae Silesianae (Vol. 9, pp. 29-32).
Kotrys, D. (2012). Hermite–Hadamard inequality for convex stochastic processes. Aequationes Mathematicae, 83(1), 143-151.
Kotrys, D. (2013). Remarks on strongly convex stochastic processes. Aequationes Mathematicae, 86(1), 91-98.
Barráez, D., González, L., Merentes, N., & Moros, A. (2015). On h-convex stochastic processes. Mathematica Aeterna, 5(4), 571-581.
Zhou, H., Saleem, M. S., Ghafoor, M., & Li, J. (2020). Generalization of h‐Convex Stochastic Processes and Some Classical Inequalities. Mathematical Problems in Engineering, 2020(1), 1583807.
Okur, N., Iscan, I., & Dizdar, E. Y. (2019). Hermite-Hadamard type inequalities for p-convex stochastic processes. An International Journal of Optimization and Control, 9(2), 148-153.
Maden, S., Tomar, M., & Set, E. (2015). Hermite-Hadamard type inequalities for s-convex stochastic processes in first sense. Pure and Applied Mathematics Letters, 2015, 1-7.
Set, E., Tomar, M., & Maden, S. (2014). Hermite-Hadamard type inequalities for s-convex stochastic processes in the second sense. Turkish Journal of Analysis and Number Theory, 2(6), 202-207.
Akdemir, H. G., Bekar, N. O., & Iscan, İ. (2014). On preinvexity for stochastic processes. Istatistik Journal of the Turkish Statistical Association, 7(1), 15-22.
Özcan, S. (2019). Hermite-Hadamard type inequalities for exponentially p-convex stochastic processes. Sakarya University Journal of Science, 23(5), 1012-1018.
Ozcan, S. (2019). Hermite-Hadamard type inequalities for m-convex and \((\alpha, m)\)-convex stochastic processes. International Journal of Analysis and Applications, 17(5), 793-802.
Fu, H., Saleem, M. S., Nazeer, W., Ghafoor, M., & Li, P. (2021). On Hermite-Hadamard type inequalities for n-polynomial convex stochastic processes. Aims Math, 6(6), 6322-6339.
Zine, H., & Torres, D. F. (2020). A stochastic fractional calculus with applications to variational principles. Fractal and Fractional, 4(3), 38.
Chen, P., Quarteroni, A., & Rozza, G. (2013). Stochastic optimal Robin boundary control problems of advection-dominated elliptic equations. SIAM Journal on Numerical Analysis, 51(5), 2700-2722.
Cuoco, D. (1997). Optimal consumption and equilibrium prices with portfolio constraints and stochastic income. Journal of Economic Theory, 72(1), 33-73.
Cvitanić, J., & Karatzas, I. (1992). Convex duality in constrained portfolio optimization. The Annals of Applied Probability, 767-818.
Xu, Y., & Yin, W. (2015). Block stochastic gradient iteration for convex and nonconvex optimization. Siam Journal on Optimization, 25(3), 1686-1716.
Sobczyk, K. (2013). Stochastic Differential Equations: With Applications to Physics and Engineering (Vol. 40). Springer Science & Business Media.
Afzal, W., Aloraini, N. M., Abbas, M., Ro, J. S., & Zaagan, A. A. (2024). Some novel Kulisch-Miranker type inclusions for a generalized class of Godunova-Levin stochastic processes. Aims Mathematics, 9(2), 5122-5146.
Afzal, W., Botmart, T., Afzal, W., & Botmart, T. (2023). Some novel estimates of Jensen and Hermite-Hadamard inequalities for h-Godunova-Levin stochastic processes. Aims Math, 8(3), 7277-7291.
Afzal, W., Prosviryakov, E. Y., El-Deeb, S. M., & Almalki, Y. (2023). Some new estimates of hermite–hadamard, ostrowski and jensen-type inclusions for h-convex stochastic process via interval-valued functions. Symmetry, 15(4), 831.
Afzal, W., Abbas, M., Eldin, S. M., & Khan, Z. A. (2023). Some well known inequalities for (h1, h2)-convex stochastic process via interval set inclusion relation. AIMS Mathematics, 8(9), 19913–19932.
Abbas, M., Afzal, W., Botmart, T., & Galal, A. M. (2023). Jensen, Ostrowski and Hermite-Hadamard type inequalities for h-convex stochastic processes by means of center-radius order relation. Aims Mathematics, 8(7), 16013-16030.
Feng, B., Ghafoor, M., Chu, Y. M., Qureshi, M. I., Xue, F., Chuang, Y., & Qiao, X. (2020). Hermite-Hadamard and Jensen’s type inequalities for modified (p, h)-convex functions. Aims Mathematics, 5(6), 6959.
Afzal, W., Eldin, S. M., Nazeer, W., Galal, A. M., Afzal, W., Eldin, S. M., … & Galal, A. M. (2023). Some integral inequalities for harmonical cr-h-Godunova–Levin stochastic processes. Aims Math, 8, 13473-13491.
Okur, N., Iscan, I., & Dizdar, E. Y. (2019). Hermite-Hadamard type inequalities for p-convex stochastic processes. An International Journal of Optimization and Control, 9(2), 148-153.
El-Achky, J., & Taoufiki, S. (2022). On \((p; h)\)–convex stochastic process. Journal of Interdisciplinary Mathematics, 25(4), 1105-1116.
Gonzales, L., Materano, J., & Lopez, M. V. (2016). Ostrowski-type inequalities via h-convex stochastic processes. Japanese Journal of Mathematics, 16(2), 15-29.
Anderson, D. R., & Ulness, D. J. (2015). Properties of the Katugampola fractional derivative with potential application in quantum mechanics. Journal of Mathematical Physics, 56(6), 063502.
