Hermite-Hadamard, Jensen, and Fractional Integral Inequalities for Generalized -Convex Stochastic Processes

Abstract

The stochastic process is one of the important branches of probability theory which deals with probabilistic models that evolve over time. It starts with probability postulates and includes a captivating arrangement of conclusions from those postulates. In probability theory, a convex function applied on the expected value of a random variable is always bounded above by the expected value of the convex function of that random variable. The purpose of this note is to introduce the class of generalized -convex stochastic processes. Some well-known results of generalized -convex functions such as Hermite-Hadamard, Jensen, and fractional integral inequalities are extended for generalized -stochastic convexity.

1. Introduction

A stochastic process is a mathematical tool commonly defined as a set of random variables in various fields of probability. Verifiably, random variables were related to or listed by a lot of numbers, normally as focuses in time, giving the translation of a stochastic process, speaking to numerical estimations, some systems randomly changing over time, such as the growth of bacterial populations, fluctuations in electrical flow due to thermal noise, or the production of gas molecules. Stochastic systems are commonly used as scientific models of systems that tend to alter in an arbitrary manner. They have applications in various fields, especially in sciences, for instance, chemistry, physics, biology, neuroscience, and ecology, in addition to technology and engineering fields, for example, picture preparing, cryptography, signal processing, telecommunications, PC science, and data theory. Furthermore, apparently, arbitrary changes in money-related markets have inspired the broad utilization of stochastic processes in fund.

Convex stochastic processes and their applications have a fundamental significance in mathematics and in probability. Nikodem [1] in 1980 proposed the idea of convex stochastic processes in his article. In 1992, -convex and Jensen-convex stochastic processes were initiated by Skowronski [2]. More recently, Kotrys presented in [3] the results on convex stochastic processes.

For more details, refer to [4]. Many studies in the literature have been performed on some extensions of convex stochastic processes and on Hermite-Hadamard type inequalities for these extensions [5].

In the present note, we purpose to investigate the idea of generalized p-convex stochastic processes. The notion of inequality as convexity has a significant place in literature [6], as it yields a broader setting in order to investigate the mathematical programming and optimization problems. Therefore, Schur type, Hermite-Hadamard, Jensen, and fractional integral inequalities and some important results for the above said processes will be obtained in this study.

We start by definition of the stochastic process [7].

Definition 1 (see [3]). Assume a probability space . A random variable is the function if is -measurable; whereas, a stochastic process is the function if is a random variable for every .
Let us review some basic notions about stochastic processes.

Definition 2 (see [3]). The stochastic process is as follows:(1)Continuous on , if for all , where represents the limit in the probability;(2)Mean-square continuous on , if for every , Where represents an expectation of the random variable . It is obvious in probability that if a stochastic process is mean-square continuous, then it is also continuous, but the converse is not true;(3)Mean square is differentiable at , if there is a random variable , such that for every ,

Definition 3 (see [3]). Consider , a stochastic process with . We say a random variable to be the mean-square integral of the process on if for each normal sequence of partitions of , , and for all , we haveThen, is written asFor more on mean-square integrable stochastic processes, refer [8].

Theorem 1 (see [3]). Let us consider the Jensen-convex stochastic process that is mean-square continuous on ; then, we havefor all , . The above inequality is Hermite-Hadamard inequality for stochastic convexity.

Let us present some important generalizations of convex stochastic processes.

Definition 4 (see [9]). A stochastic process is said to be generalized convex if for and ,

Definition 5 (see [10]). An interval is a p-convex set if for all , , and p = 2m + 1 or , r = 2s + 1, n = 2t + 1, and .

Definition 6 (see [10]). A function is -convex, if for and , we havewhere is a -convex set.

Remark 1 (see [11]). If be a real interval and , thenAccording to Remark 1, we can give a different version of the definition of -convex function as follows:

Definition 7 (see [11]). If be a real interval and . A function is said to be a -convex function iffor all and . If the inequality (10) is reversed, then is said to be -concave.

Definition 8 (see [12]). A process , where is a -convex set, is said to be a -convex stochastic process if for and , we haveIn [13], the following functions are defined.

Definition 9.

Definition 10. For , ,Now, we are in position to define the main notion of this article.

Definition 11. A stochastic process is said to be generalized -convex, if for and , we haveIf the inequality in (14) is reversed, then is the generalized -concave.

Remark 2. It is obvious that the inequality (14) reduces to the convex stochastic process for and .

Example 1. Consider a stochastic process defined by , , and ; then, is the generalized -convex.
We organize our study as follows. First we derive some basic properties for this generalization. In next section, Schur type inequality is obtained. The third, fourth, and fifth sections are devoted to Hermite-Hadamard, Jensen, and fractional integral inequalities for generalized -convex stochastic processes.

Proposition 1. Let be two generalized -convex stochastic processes:(1)If is additive, then is also a generalized -convex stochastic process(2)If is nonnegatively homogeneous, then , for any , is the generalized -convex stochastic process

The proof Proposition 1 is straightforward.

Theorem 2. Assume a nonempty collection of generalized -convex stochastic processes, such that(1)There exist and , such that for all (2)For each exists in ; then, the stochastic process defined by for all is the generalized -convex.

Proof. For any and , we havewhich is as required.

2. Schur Type Inequality

Theorem 3. For and , let is the generalized -convex stochastic process. Then, , such that and , and we have

Proof. Let be given. Then, we can easily see thatSetting , . As is generalized -convex, soBy assuming and multiplying the above inequality by , we get inequality (16).

3. Hermite-Hadamard Type Inequality

Theorem 4. For and , let a mean-square generalized -convex stochastic process , which is integrable. Then, for any , , the following inequality holds almost everywhere:

Proof. Take and ; so,Since is the generalized -convex, so we haveIntegrating w.r.t “,” the above inequality on [0, 1],which impliesNow,which impliesSimilarly,Adding (25) and (26),Combining (23) and (27), we obtain the inequality (22).

Remark 3. For and in (22), we get Hermite-Hadamard inequality (6) for the convex stochastic process.

4. Jensen Type Inequality

The following result will be helpful in the derivation of Jensen’s type inequality for the generalized -convex stochastic process.

Lemma 1. Let be the positive real numbers . Assume be a generalized -convex stochastic process and ; then, we have almost everywherewhere .

Theorem 5 (Jensen type inequality). Let be a generalized -convex stochastic process and be nondecreasing, nonnegatively sublinear in the first variable; then, we have almost everywherewhere and and .

Proof. Since is nondecreasing, nonnegatively sublinear in the first variable, so using Lemma 1, we get

5. Fractional Integral Inequalities

Lemma 2 (see [12]). Assume a stochastic process which is mean-square differentiable on and with . If , then we have almost everywhere

Theorem 6. For and and under the assumptions of Lemma 2 with , a generalized -convex stochastic process on for , then we have almost everywherewhere

Proof. By making use of Lemma 2 and power mean-integral inequality, we haveHence, by generalized -convexity of on , we haveIt is easy to check that

Remark 4. By setting in (32), we get Theorem 4 of [12].
We will get the following Corollary by taking in (32).

Corollary 1. If is generalized -convex on , then we have almost everywherewhere and are defined in Theorem 6.

Remark 5. If we take in (37), then we have Corollary 4 of [12].

Theorem 7. For and and under the assumptions of Lemma 2 with , a generalized -convex stochastic process on for , , then we have almost everywherewhere

Proof. By making use of Lemma 2, Holder’s inequality, and generalized -convexity of on , we havewhere an easy calculation givesSubstituting equations (41) and (42) into (38), the proof is completed.

Remark 6. By taking in Theorem 7, then we obtain Theorem 6 of [12].

Theorem 8. For and and under the assumptions of Lemma 2 with , a generalized -convex on for , , then we have almost everywherewhere

Proof. From Lemma 2, Holder’s inequality, and generalized -convexity of on , we havewhere an easy calculation givesSubstituting (46) and (47) into (43), we obtain the required result.

6. Conclusion

There are many applications of stochastic processes, for instance, the Kolmogorov–Smirnoff test on equality of distributions. The other application includes sequential analysis and quickest detection. In this study, we have presented a new class of convex stochastic processes which are generalized p-convex and established Jensen, Hermite-Hadamard, and fractional integral inequalities for this class. Our conclusions are applicable, since the expected value of a random variable is consistently bounded above by the expected value of the convex function of that random variable. It will be interesting to find parallel results by using the proposed definition in this study in the setting of other fractional integrals [14, 15].

Data Availability

The data used to support the findings of this study are included within this article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

Fangfang Ma validated the results, prepared the final draft of the manuscript, and arranged the funding for this study. Waqas Nazeer proved the main results. Mamoona Ghafoor wrote the first draft of the manuscript.

Acknowledgments

The study was supported by Science and Technology Nova Plan of Beijing City (Z201100006820122) and Fundamental Research Funds for the Central Universities (2020RC14).